mm^ 


,<,</'<'(  1  '  ,* ' 


1 


7^  LIBRARY 

OF  THE 

University  of  California. 


? 


GIFT    OF 


..\).  ^...Uw...  L^ 


Class 


,/...••   •-..    ...^.      K       ^.N,.      .     ^ 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/essentialsofalgeOOwellrich 


^- 


WELLS'    MATHEMATICAL   SERIES. 


Academic  Arithmetic. 

The  Essentials  of  Algebra. 

Academic  Algebra. 

Higher  Algebra. 

University  Algebra. 

College  Algebra. 

Plane  Geometry.     Revised. 

Solid  Geometry.     Revised. 

Plane  and  Solid  Geometry.     Revised. 

Essentials  of  Geometry.  Plane. 

Essentials  of   Geometry.  Solid. 

Essentials  of  Geometry.  Plane  and  Solid. 

New  Plane  and  Spherical  Trigonometry. 

Plane  Trigonometry. 

Essentials  of  Trigonometry. 

Logarithms  (flexible  covers). 


Special  Catalogue  and  Terms  on  application. 


>h- 


ESSENTIALS    OF   ALGEBRA 


FOR 


SECONDARY   SCHOOLS 


BY 


WEBSTER   WELLS,   S.B. 

PROFESSOR  OF  MATHEMATICS  IN  THE  MASSACHUSETTS 
INSTITUTE  OF  TECHNOLOGY 


VERSITY 

OF 


LEACH,   SHEWELL   AND  COMPANY 

NEW  YORK      BOSTON      CHICAGO 


Copyright,  1897, 
By  WEB8TEE   WELLS. 


NorhJooU  53resg 

J.  S.  Cushing  &  Co.  -  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 


The  cordial  reception  which  the  author's  other  Algebras 
have  received  at  the  liands  of  the  educational  public,  their 
extensive  use  in  schools  of  the  highest  rank  in  all  parts  of 
the  country,  the  appreciative  reconnnendations  which  have 
come  to  him  from  instructors  of  reputation,  lead  him  to 
believe  that  this  latest  attempt  to  adequately  meet  the 
demands  of  the  best  secondary  schools  will  be  cordially 
welcomed. 

Our  teachers  are  progressive,  and  the  author  who  fails  to 
keep  abreast  of  the  times,  and  in  sympathy  with  the  best 
educational  thought  and  methods,  will  appeal  in  vain  for 
the  patronage  and  sympathies  of  his  fellow-teachers. 

Fully  conscious  of  the  above  truth,  the  author  earnestly 
recommends  "  The  Essentials  of  Algebra  "  to  the  attention 
of  the  educational  public. 

It  affords  a  thorough  and  complete  treatment  of  elemen- 
tary Algebra,  and  attention  is  especially  invited  to  the  fol- 
lowing features  :  — 

The  introduction  of  easy  problems  at  the  very  outset ;  §  5. 

The  Addition  and  Multiplication  of  Positive  and  Nega- 
tive Numbers  ;  §§  14  to  19. 

The  Addition  of  Similar  Terms  ;  §  31. 

The  discussion  of  Simple  Equations,  not  involving  Frac- 
tions, directly  after  Division  ;  Chap.  VII. 

The  suggestions  in  regard  to  the  solution  of  problems ; 
§§  76,  77. 

The  discussion  of  the  theoretical  principles  involved  in 
the  handling  of  fractions  ;  §§  129,  136,  143,  145. 


184017 


iv  PllEFACE. 

The  examples  on  page  176. 

The  discussion  of  square  roots  and  cube  roots  of  arith- 
metical numbers;  §§  197,  198,  203,  204. 

The  examples  at  the  end  of  §  229. 

The  solution  of  equations  by  factoring  ;  §§  266,  267. 

The  factoring  of  a  quadratic  expression  when  the  co- 
efficient of  fl^  is  a  perfect  square  ;  §  286. 

Great  care  has  been  taken  to  state  the  various  definitions 
and  rules  with  accuracy,  and  every  principle  has  been  dem- 
onstrated with  strict  regard  to  the  logical  principles  in- 
volved. As  a  rule,  no  definition  has  been  introduced  until 
its  use  became  necessary. 

The  examples  and  problems  have  been  selected  with  great 
care,  are  ample  in  number,  and  thoroughly  graded.  They 
are  especially  numerous  in  the  important  chapters  on  Fac- 
toring, Fractions,  and  Radicals. 

The  latest  English  practice  has  been  followed  in  writing 
Arithmetic,  Geometric,  and  Harmonic,  for  Arithmetical, 
Geometrical,  and  Harmonical,  in  the  progressions. 

The  author  wishes  to  acknowledge,  with  hearty  thanks, 
the  many  suggestions  and  the  assistance  that  he  has  received 
from  principals  and  teachers  of  secondary  schools  in  all 
parts  of  the  country,  in  improving  and  perfecting  the 
work. 

WEBSTER  WELLS. 

Massachusetts  Institute  of  Technology, 
March,  1897. 


CONTENTS. 


PAGE 

I.    Definitions  and  Notation 1 

Solution  of  Problems  by  Algebraic  Methods    ....  2 

Algebraic  Expressions 6 

II.    Positive  and  Negative  Numbers 9 

Addition  of  Positive  and  Negative  Numbers    ....  11 

Multiplication  of  Positive  and  Negative  Numbers     .     .  12 

III.  Addition  and  Subtraction  of  Algebraic  Expressions  .  15 

Addition  of  Monomials 16 

Addition  of  Polynomials 20 

Subtraction 21 

Subtraction  of  Monomials 22 

Subtraction  of  Polynomials 23 

IV.  Parentheses 26 

Removal  of  Parentheses 26 

Introduction  of  Parentheses 28 

V.    Multiplication 29 

Multiplication  of  Monomials 30 

Multiplication  of  Polynomials  by  Monomials  ....  32 

Multiplication  of  Polynomials  by  Polynomials    ...  32 

VI.   Division 37 

Division  of  Monomials 38 

Division  of  Polynomials  by  Monomials 39 

Division  of  Polynomials  by  Polynomials 40 

VII.    Simple  Equations 48 

Properties  of  Equations 49 

Solution  of  Simple  Equations 50 

Problems 52 

VIII.    Important  Rules  in  Multiplication  and  Division  .     .  59 

IX.    Factoring 67 

V 


vi  CONTENTS. 

PAGE 

X.    Highest  Common  Factor 81 

XI.   Lowest  Common  Multiple 91 

XII.    Fractions 96 

Reduction  of  Fractions 97 

Addition  and  Subtraction  of  Fractions 105 

Multiplication  of  Fractions Ill 

Division  of  Fractions 113 

Complex  Fractions 115 

.  XIII.    Simple  Equations  (Continued) 120 

Solution  of  Equations  containing  Fractions      .     .     .  120 

Solution'of  Literal  Equations 124 

Solution  of  Equations  involving  Decimals    ....  12(j 

Problems 127 

Problems  involving  Literal  Equations 136 

XIV.   Simultaneous  Equations 

Containing  Two  Unknown  Quantities     .....  138 

XV.    Simultaneous  Equations 

Containing  more  than  Two  Unknown  Quantities .     .  160 

XVI.    Problems 

Involving  Simultaneous  Equations 154 

XVII.    Inequalities 165 

XVIII.    Involution     .  * 170 

Involution  of  Monomials 170 

Square  of  a  Polynomial 171 

Cube  of  a  Binomial 172 

—  XIX.    Evolution 174 

Evolution  of  Monomials 174 

Square  Root  of  a  Polynomial 176 

Square  Root  of  an  Arithmetical  Number      .     .    ,     .  179 

Cube  Root  of  a  Polynomial 183 

Cube  Root  of  an  Arithmetical  Number 186 

XX.    Theory  of  Exponents 191 

XXI.   Radicals 201 

Reduction  of  a  Radical  to  its  Simplest  Form    .     .     .  201 

Addition  and  Subtraction  of  Radicals 205 

To  Reduce  Radicals  of  Different  Degrees  to  Equiva- 
lent Radicals  of  the  Same  Degree 206 


CONTENTS. 


vil 


XXI.   Radicals  (Continued).  page 

Multiplication  of  Radicals 207 

Division  of  Radicals 210 

Involution  of  Radicals 212 

Evolution  of  Radicals 212 

To  Reduce  a  Fraction  having  an  Irrational  Denom- 
inator to  an  Equivalent  Fraction  whose  Denom- 
inator is  Rational 213 

Properties  of  Quadratic  Surds 215 

Imaginary  Numbers 218 

Solution  of  Equations  containing  Radicals    .  .  222 

XXII.    Quadratic  Equations 224 

Pure  Quadratic  Equations 224 

Affected  Quadratic  Equations 226 

Problems 238 

XXIII.  Equations  Solved  like  Quadratics 243 

Equations  in  the  Quadratic  Form 243 

XXIV.  Simultaneous  Equations 

Involving  Quadratics 248 

Problems 258 

**Jlp  XXV.   Theory  or  Quadratic  Equations 261 

'  Factoring .  263 

Discussion  of  the  General  Equation 268 

XXVI.    Zero  and  Infinity 270 

Variables  and  Limits 270 

The  Problem  of  the  Couriers 272 

XXVII.   Indeterminate  Equations 275 

XXVIII.   Ratio  and  Proportion 278 

Properties  of  Proportions  .     . ' 279 

XXIX.    Variation 287 

-^XXX.   Progressions    . 291 

Arithmetic  Progression 291 

Geometric  Progression 299 

Harmonic  Progression 307 

-AZ^XXI.    The  Binomial  Theorem 310  ^^ 

^'  Positive  Integral  Exponent 310  *' 


viii  CONTENTS. 

PAGE 

XXXII.    Undetermined  Coefficients 317 

Convergency  and  Divergency  of  Series     .     .     .     .  317 

The  Theorem  of  Undetermined  Coefficients  .     .     .  320 

Expansion  of  Fractions  into  Series 321 

Expansion  of  Radicals  into  Series 323 

Partial  Fractions 324 

Reversion  of  Series 330 

XXXIII.  The  Binomial  Theorem 332 

Fractional  and  Negative  Exponents     ....  332 

XXXIV.  Logarithms .339 

Properties  of  Logarithms 341 

Use  of  the  Table 346 

Applications 351 

Arithmetical  Complement 353 

Exponential  Equations 357 


Answers  to  the  Examples. 


ALGEBRA, 


I.    DEFINITIONS  AND  NOTATION. 

1.  Ill  Algebra,  the  operations  of  Arithmetic  are  abridged 
and  generalized  by  means  of  Symbols. 

2.  Symbols  which  represent  Numbers. 

The  symbols  generally  employed  to  represent  numbers 
are  the  figures  of  Arithmetic  and  the  letters  of  the  Alphabet. 

Known  Numbers  are  usually  represented  by  the  first 
letters  of  the  alphabet,  as  a,  6,  c. 

Unknown  Numbers,  or  those  whose  values  are  to  be 
determined,  are  usually  represented  by  the  last  letters 
of  the  alphabet,  as  a?,  y,  z. 

3.  Symbols  which  represent  Operations. 

The  following  symbols  have  the  same  meaning  in  Alge- 
bra as  in  Arithmetic : 

+,  read  ^^plusy 

—  ,  read  ^'^ninus.'^- 

X ,  read  "  times^"  "  mto,"  or  "  multiplied  by^ 

-!-,  read  ^'divided  by.^^ 

The  sign  of  multiplication  is  usually  omitted  in  Algebra, 
except  between  arithmetical  figures. 

Thiis,  2  X  a;  is  written  2  x. 

Division  is  usually  indicated  by  a  horizontal  line. 

Thus,  a -J- 6  is  written  -• 
b 


2  ALGEBRA. 

4.  The  Sign  of  Equality,  =,  is  read  ^'equalsj^'  or  'Hs 
equal  toJ^ 

An  Equation  is  a  statement  that  two  numbers  are  equal. 

SOLUTION  OF  PROBLEMS  BY  ALGEBRAIC  METHODS. 

5.  The  following  examples  will  illustrate  the  use  of 
Algebraic  symbols  in  the  solution  of  problems. 

The  utility  of  the  process  consists  in  the  fact  that  the 
unknown  numbers  are  represented  by  symbols,  and  that 
the  various  operations  are  stated  in  Algebraic  language. 

1.  The  sum  of  two  numbers  is  30,  and  the  greater  exceeds 
the  less  by  4 ;  what  are  the  numbers  ? 

We  will  represent  the  less  number  by  x. 
Then  the  greater  will  be  represented  by  x  +  4- 
By  the  conditions  of  the  problem,  the  sum  of  the  greater  number 
and  the  less  is  30  ;  this  is  stated  in  Algebraic  language  as  follows : 

a;  +  4  +  X  =  30,  (1) 

But  the  sum  of  x  and  x  is  twice  x,  or  2x;  whence,  equation  (1) 
may  be  written 

2  x  +  4  =  30. 

Now  it2x  plus  4  equals  30,  2  x  must  equal  30  -  4,  or  26. 

Whence,  2x  =  26.  • 

But  if  twice  x  is  26,  x  must  be  one-half  of  26,  or  13. 

Hence,  the  less  number  is  13,  and  the  greater  is  13  +  4,  or  17. 

The  written  work  will  stand  as  follows : 

Let  X  =  the  less  number. 

Then,  x  +  4  =  the  greater  number. 

By  the  conditions,  x  -\-  4  +  x  =  SO. 

Or,  2x  +  4  =  30. 

Whence,  2  x  =  26. 

Dividing  by  2,  x  =  13,  the  less  number. 

Whence,  x  +  4  =  17,  the  greater  number. 


DEFINITIONS  AND  NOTATION.  3 

2.  The  sum  of  the  ages  of  A  and  B  is  109  years,  and  A 
is  13  years  younger  than  B ;  find  their  ages> 

Let  X  represent  the  number  of  years  in  B's  age. 
Then,  x  —  13  will  represent  the  number  of  years  in  A's  age. 
By  the  conditions  of  the  problem,  the  sum  of  the  ages  of  A  and  B 
is  109  years. 

Whence,  a;  +  x  -  13  =  109. 

Or,  2  X  -  13  =  109. 

Now  if  2  X  minus  13  equals  109,  2  x  must  equal  109  +  13,  or  122. 

Whence,  2  x  =  122. 

Dividing  by  2,  x  =  61,  the  number  of  years  in  B's  age. 

And,  X  —  13  =  48,  the  number  of  years  in  A's  age. 

The  written  work  will  stand  as  follows : 

Let  X  =  the  number  of  years  in  B's  age. 

Then,  x  —  13  =  the  number  of  years  in  A's  age. 

By  the  conditions,  x  +  x  —  13  =  109. 

Or,  2x-13  =  109. 

Whence,  2  x  =  122. 

Dividing  by  2,  x  =  61,  the  number  of  years  in  B's  age. 

Therefore,  x  —  13  =  48,  the  number  of  years  in  A's  age. 

3.  A,  B,  and  C  together  have  %^^.  A  has  one-half  as 
much  as  B,  and  C  has  3  times  as  much  as  A.  How  much 
has  each  ? 

Let  X  =  the  number  of  dollars  A  has. 

Then,  2  x  =  the  number  of  dollars  B  has, 

and  3  X  =  the  number  of  dollars  C  has. 

By  the  conditions, 

x  +  2x  +  3x  =  66. 

But  the  sum  of  x,  twice  x,  and  3  times  x  is  6  times  x,  or  6  x. 

Whence,  6x  =  66. 

Dividing  by  6,  x  =  11,  the  number  of  dollars  A  has. 

Whence,  2  x  =  22,  the  number  of  dollars  B  has, 

and  3  X  =  33,  the  number  of  dollars  C  has. 


ALGEBRA. 


PROBLEMS. 


4.  The  greater  of  two  numbers  is  4  times  the  less,  and 
their  sum  is  70.     What  are  the  numbers  ? 

5.  Tlie  sum  of  the  ages  of  A  and  B  is  116  years,  and  A 
is  18  years  younger  than  B.     What  are  their  ages  ? 

6.  Divide  123  into  two  parts,  such  that  the  greater 
exceeds  the  less  by  67. 

7.  The  sum  of  the  ages  of  A  and  B  is  102  years,  and  A 
is  26  years  older  than  B.     What  are  their  ages  ? 

8.  Divide  $  93  between  A  and  B,  so  that  A  may  receive 
$  23  less  than  B. 

9.  Divide  $  56  between  A  and  B,  so  that  A  may  receive 
6  times  as  much  as  B. 

10.  Divide  85  into  two  parts,  one  of  which  shall  be  19 
less  than  the  other. 

11.  Divide  $  72  between  A  and  B,  so  that  A  may  receive 
one-third  as  much  as  B. 

12.  A  certain  hall  contains  425  persons ;  there  are  3  times 
as  many  men  as  women,  and  4  times  as  many  women  as 
children.     How  many  are  there  of  each? 

13.  A  man  had  ^4.95.  After  spending  a  certain  sum, 
he  found  that  he  had  left  4  times  as  much  as  he  had  spent. 
How  much  did  he  spend? 

14.  A,  B,  and  C  together  have  $96.  B  has  twice  as 
much  money  as  C,  and  A  has  as  much  as  B  and  C  together. 
How  much  has  each  ? 

15.  The  sum  of  three  numbers  is  168.  The  second  is  23 
less  than  the  first,  and  the  third  is  3  times  the  second. 
What  are  the  numbers  ? 

16.  A,  B,  and  C  together  have  $  230.  A  has  $  21  more 
than  B,  and  $  17  less  than  C.     How  much  has  each  ? 


DEFINITIONS   AND   NOTATION.  5 

17.  A  watch  and  chain  are  together  worth  ^5G,  and  tlie 
chain  is  worth  one-sixth  as  much  as  the  watch.  What  is 
the  value  of  each  ? 

18.  Divide  169  into  three  parts,  the  first  of  which  is  one- 
half  of  the  second,  and  the  second  one-tifth  of  the  third. 

19.  Divide  $  144  into  three  parts  such  that  the  second 
is  one-third  of  the  first,  and  one-fourth  of  the  third. 

20.  A  man  bought  a  cow,  a  sheep,  and  a  hog  for  ^.84. 
The  price  of  the  hog  was  one-fifth  the  price  of  the  cow,  and 
$  7  less  than  the  price  of  the  sheep.  What  was  the  price 
of  each  ? 

21.  The  sum  of  three  numbers  is  127.  The  first  is  one- 
half  of  the  third,'  and  17,  greater  than  the  second.  What 
are  the  numbers  ? 

22.  At  a  certain  election,  two  candidates,  A  and  B,  to- 
gether received  508  votes;  and  A  had  a  majority  of  136. 
How  many  did  each  receive  ? 

23.  The  sum  of  the  ages  of  A,  B,  and  C  is  101  years. 
A  is  17  years  younger  than  B,  and  15  years  older  than  C. 
What  are  their  ages  ? 

24.  Divide  $  174  between  A,  B,  and  C,  so  that  A  may 
receive  4  times  as  much  as  B,  and  $  42  more  than  C. 

26.  My  horse,  carriage,  and  harness  are  together  worth 
$  456.  The  carriage  is  worth  8  times  as  much  as  the  har- 
ness, and  $  48  less  than  the  horse.     Find  the  value  of  each. 

26.  Divide  $  155  into  three  parts  such  that  the  first  shall 
be  5  times  the  second,  and  one-fifth  of  the  third. 

27.  At  a  certain  election,  three  candidates.  A,  B,  and  C, 
together  received  512  votes.  A  received  28  less  than  B, 
and  64  less  than  C.     How  many  did  each  receive  ? 

28.  Divide  $  69  between  A,  B,  C,  and  D,  so  that  A  may 
receive  ^5  more  than  B,  C  as  much  as  A  and  B  together, 
and  D  as  much  as  A  and  C  together. 


6  ALGEBRA. 

29.  The  sum  of  four  numbers  is  160.  The  first  is  3 
times  the  second,  the  second  3  times  the  third,  and  the 
third  3  times  the  fourth.     What  are  the  numbers  ? 

DEFINITIONS. 

6.  If  a  number  be  multiplied  by  itself  any  number  of 
times,  the  result  is  called  a  power  of  that  number. 

An  Exponent  is  a  number  written  at  the  right  of,  and 
above  another  number,  to  indicate  what  power  of  the  latter 
is  to  be  taken. 

Thus, 
o?,  read  ''  a  square/^  or  "  a  second  power/'  denotes  a  x  a ; 
a^,  read  "  a  cube,''  or  "  a  third  power,"  denotes  a  x  a  x  a; 
a*,  read  "  a  fourth,"  or  "  a  fourth  power,"  denotes  axaxaxa, 
and  so  on. 

If  no  exponent  is  expressed,  the  first  power  is  understood. 

Thus,  a  is  the  same  as  a^. 

7.  Symbols  of  Aggregation. 

The  parentheses  (  ),  the  brackets  [  ],  the  braces  {.  \ ,  and 

the  vinculum ,  indicate  that  the  numbers  enclosed  by 

them  are  to  be  taken  collectively ;  thus, 


(a  -\-b)  X  c,  \_a-\-b~\x  c,   \a-\-b\x  c,  and  a  -f-  6  x  c 

all  indicate  that  the  result  obtained   by  adding  6  to  a  is 
to  be  multiplied  by  c. 

ALGEBRAIC  EXPRESSIONS. 

8.  An  Algebraic  Expression,  or  simply  an  Expression,  is 
a  number  expressed  in  algebraic  symbols ;  as, 

2,  a,  or  2x^-^ab-{-5. 

The  Numerical  Value  of  an  expression  is  the  result 
obtained  by  substituting  particular  numerical  values  for 
the  letters  involved  in  it,  and  performing  the  operations 
indicated. 


DEFINITIONS  AND   NOTATION 


It'  Find  the  numerical  value  of  the  expression 

when  a  =  4,  b  =  S,  c  =  5,  and  d  =  2. 

We  have,  4a  +  ^-rf3  =  4x4+  ^-^  -  23 

b  •  3 

=  16  +  10  -  8  =  18,  Ans. 


EXAMPLES. 

Find  the  numerical  value  of  each  of  the  following  when 
=  3,  b  =  o,  c  =  2,  d  =  4,  m  =  4,  and  7i  =  3 : 

8.  a'^d''. 

9.  ^  +  5-1 
be     a 

10.   Sa'-9(f. 

11.  i+ui+i. 

a      6>      c     tt 

12    ^  —  .^ 
'   2c2     4^2' 

13.   ^V^-  +  ^. 
36"  •  62^c2^f/2 

If  the   expression   involves  parentheses,   the   operations 
indicated  ivithin  the  parentheses  must  be  performed  first. 

14.    Find  the   numerical  value,  when  a  =  9,  6  =  7,  and 
c  =  4,  of 

{a-h)(b  +  c)-pdi. 
0  —  c 

We  have,  a  -  6  =  2,  6  +  c  =  11,  a  +  6  =  16,  and  Z>  -  c  =  3. 
Then  the  numerical  value  of  the  expression  is 

2xll-l«  =  22_l«  =  50,  ^„,. 
3  3       3 


2. 

acZ^  -  b(^. 

3. 

Sabcd. 

4. 

4  a^fZ  -  5  6c  - 

-6cd. 

5. 

ab  _^cd 
To" 

6. 

a     6      c 

7. 

5a- 

g  .  ALGEBRA. 

Find  the  numerical  value  of  each  of  the  following  expres- 
sions, when  a=  Of  b  =  3,  c  =  4,  and  d  =  2  : 

1-5.    f^-ff-  17.   Sa\c-d)-2b\c-{-d). 

16.  (a'-{-b'-cy.  18.  5(a  +  by -9(c- ay. 

19.  (a  -  5)  (6  +  c)  -  (c  -  d)  {d  +  a). 

20.  (a  -  6  +  c  -  d)  (a  +  6  -  c  -  d). 

21     3a  —  56  +  7  c  go    ^  +  6      6  +  c      c-fc/ 

'4a  —  36H-6c  b  -{-  c     c.-\-  d     d  -\-  a 

Find  the  numerical  value  of  each  of  the  following  expres- 
sions, when  a  =  -J,  6  =  ^,  c  =  |,  and  a;  =  3 : 

.  23    g  +  6  _  a  —  b  24     16  a  —  18  6  +  15  c 

j      '   a-6     a +  6*  "    44 a -32 6 -27c" 

25.   a^-(4a-5c)a^+(2a  +  6)a;-12a6c. 


\6      ay  \a      6     cj 


9.  Axioms. 

An  Axiom  is  a  truth  which  is  assumed  as  self-evident. 
Algebraic  operations  are  based  upon  the  following  axioms : 

1.  If  the  same  operation  be  performed  upon  equal  number Sy 
the  resulting  numbers  will  be  equal. 

2.  If  the  same  number  be  both  added  to,  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  changed, 

S.  If  a  number  be  both  multiplied  and  divided  by  another, 
the  value  of  the  former  will  not  be  changed. 

4.  Numbers  which  are  equal  to  the  same  number,  are  equal 
to  each  other. 


POSITIVE   AND   NEGATIVE   NUMBERS. 


II.     POSITIVE  AND  NEGATIVE  NUMBERS. 

10.  Many  concrete  magnitudes  are  capable  of  existing  in 
two  opposite  states. 

Thus,  in  financial  transactions,  we  may  have  gains,  or 
losses;  in  the  thermometer,  we  may  have  temperatures  above 
zero,  or  beloiv  zero ;  a  place  on  the  surface  of  the  earth  may 
be  in  north  latitude,  or  south  latitude ;  etc. 

The  signs  -f-  and  — ,  besides  indicating  the  operations  of 
addition  and  subtraction,  are  also  used  in  Algebra  to  distin- 
guish between  the  opposite  states  of  magnitudes  like  the 
above. 

Thus,  in  financial  transactions,  we  may  indicate  gains  or 
assets  by  the  sign  -f ,  and  losses  or  debts  by  the  sign  —  ;  for 
example,  the  statement  that  a  man's  property  is  —  f  100, 
means  that  he  has  debts  or  liabilities  to  the  amount  of  $  100. 

Again,  in  the  thermometer,  we  may  indicate  temperatures 
above  zero  by  the  sign  -f ,  and  temperatures  beloiv  zero  by 
the  sign  —  ;  for  example,  +  25°  means  25°  above  zero,  and 
—  10°  means  10°  below  zero. 

Also,  we  may  indicate  north  latitude  and  west  longitude 
by  the  sign  +,  and  south  latitude  and  east  longitude  by  the 
sign  —  ;  thus,  a  place  in  latitude  —  30°,  longitude  +  95°, 
would  be  in  latitude  30°  south  of  the  equator,  and  in  longi- 
tude 95°  west  of  Greenwich. 

EXERCISES. 

11.  1.  At  7  A.M.  the  temperature  is  —  13°;  at  noon  it  is 
8°  warmer,  and  at  6  p.m.  it  is  5°  colder  than  at  noon.  Re- 
quired the  temperatures  at  noon  and  at  6  p.m. 

2.  At  7  A.M.  the  temperature  is  +6°;  at  noon  it  is  14° 
colder,  and  at  6  p.m.  it  is  8°  warmer  than  at  noon.  Required 
the  temperatures  at  noon  and  at  6  p.m. 


10  ALGEBRA. 

3.  What  is  tlie  difference  in  latitude  between  two  places 
whose  latitudes  are  +  67°  and  —  48°  ? 

4.  A  man  has  bills  receivable  to  the  amount  of  $  480,  and 
bills  payable  to  the  amount  of  $  925 ;  how  much  is  he 
worth  ? 

6.  A  vessel  sails  from  the  equator  due  north  28°,  and 
then  due  south  57° ;  what  is  her  latitude  at  the  end  of  the 
voyage  ? 

6.  At  7  A.M.  the  temperature  is  —  7°,  and  at  noon  -f-  9°. 
How  many  degrees  warmer  is  it  at  nooii  than  at  7  a.m.  ? 

7.  What  is  the  difference  in  longitude  between  two  places 
whose  longitudes  are  -f  29°  and  —  86°  ? 

8.  The  temperature  at  6  a.m.  is  +14°;  and  during  the 
morning  it  grows  colder  at  the  rate  of  4°  an  hour.  Required 
the  temperatures  at  9  a.m.,  at  10  a.m.,  and  at  noon. 

12.  Positive  and  Negative  Numbers. 

If  the  pot-xtive  and  negative  states  of  any  concrete  mag- 
nitude be  expressed  without  reference  to  the  unit,  the  results 
are  called  positive  and  negative  numbers,  respectively^ 

Thus,  in  +  $  5  and  —  $  3,  +  5  is  a  positive  number,  and 
—  3  is  a  negative  number. 

For  this  reason  the  sign  +  is  called  the  positive  sign,  and 
the  sign  —  the  negative  sign. 

If  no  sign  is  expressed,  the  number  is  understood  to  be 
positive ;  thus,  5  is  the  same  as  +  5. 

The  negative  sign  can  never  be  omitted  before  a  negative 
number. 

13.  The  Absolute  Value  of  a  number  is  the  number  taken 
independently  of  the  sign  affecting  it. 

Thus,  the  absolute  value  of  —  3  is  3. 


POSITIVE   AND   NEGATIVE   NUMBERS.  H 

ADDITION  OF   POSITIVE  AND  NEGATIVE  NUMBERS. 

14.  The  result  of  Addition  is  called  the  Sum. 

We  shall  retain  for  Addition  in  Algebra  its  arithmetical 
meaning,  .so  long  as  the  numbers  to  be  added  are  positive. 

We  may  then  attach  any  meaning  we  please  to  addition 
involving  other  forms  of  number,  provided  the  new  meaning 
is  not  inconsistent  with  principles  which  have  been  pre- 
viously established. 

15.  If  a  man  gains  $  5,  and  then  loses  $  3,  he  will  be 
worth  $2. 

If  he  owes  $5,  and  then  gains  $S,  he  will  be  in  debt 
to  the  amount  of  ^2. 

If  he  owes  $5,  and  then  incurs  a  debt  of  $3,  he  will 
be  in  debt  to  the  amount  of  $S. 

Now  with  the  notation  of  §  10,  losing  $3,  or  incurring 
a  debt  of  ^  3,  may  be  regarded  as  adding  —  $  3  to  his 
property. 

Whence,    the  sum  of  +  $  5  and  -  $  3  is  -f ;^ 2 ; 
the  sum  of  -$5and  -h$3  is  -$2; 
and  the  sum  of  —  $  5  and  —  f  3  is  -  $  8. 

Or,  omitting  reference  to  the  unit, 

(+5)  +  (-3)  =  +2; 
(_5)  +  (+3)=-2; 
(_5)  +  (_3)=-8. 

We  then  have  the  following  rules : 

To  add  a  positive  and  a  negative  number,  subtract  the  less 
absolute  value  (§  13)  from  the  greater,  and  prefix  to  the  result 
the  sign  of  the  number  having  the  greater  absolute  value. 

To  add  two  negative  numbers,  add  their  absolute  values,  and 
prefix  a  negative  sign  to  the  result. 


12  ALGEBRA. 

16.  1.  Find  the  sum  of  +  10  and  —  3. 

Subtracting  3  from  10,  the  result  is  7. 
Whence,  (  +  10)  +  (  -  3)  =  +  7 ,  Ans. 

2.  Find  the  sum  of  —  12  and  +  6. 
Subtracting  6  from  12,  the  result  is  6. 
Whence,  (  -  12)  +  ( -f  6)  =  -  6,  Ans. 

3.  Add  -  9  and  -  5. 

The  sum  of  9  and  5  is  14. 

Whence,  (  -  9)  +  (  -  5)  =  -  14,  Aiis. 

EXAMPLES. 

Find  the  values  of  the  following : 

4.  (_7)  +  (-5).  10-   (-61) +  (+28). 

5.  (+9) +  (-4).  11.   (-!)  +  (+! 
6.(-8)  +  (+2,  l.(-|)  +  (-f). 

7.  (+6) +  (-15).  -       13.   (\4+\Joj).    A 

8.  (-11) +  (-16).  u.   (-18t75.)  +  (+12|).'"J* 

9.  (+52) +  (-37).  15.   (+20Jy)  +  (-13A).> 

MULTIPLICATION    OF    POSITIVE    AND    NEGATIVE 
NUMBERS. 

17.  The  terms  Multiplicand,  Multiplier,  and  Product  have 
the  same  meaning  in  Algebra  as  in  Arithmetic. 

We  shall  retain  for  Multiplication,  in  Algebra,  its  arith- 
metical meaning,  so  long  as  the  multiplier  is  a  positive 
number. 

That  is,  to  multiply  a  number  by  a  positive  integer  is  to 
add  the  first  number  as  many  times  as  there  are  units  in 
the  second. 


POSITIVE   AND   NEGATIVE   NUMBERS.  13 

For  example,  to  multiply  —  4  by  3,  we  add  —  4  three 
times. 

That  is,  (-4)  X  (4-3)  =  (-4)  +  (-4)  +(-4)  =-  12. 

We  may  then  attach  any  meaning  we  please  to  multipli- 
cation by  a  negative  number. 

18.  In  Arithmetic,  the  product  of  two  numbers  is  the 
same  in  whatever  order  they  are  taken. 

Thus,  3x5  and  5x3  are  each  equal  to  15. 

If  we  assume  this  law  to  hold  universally,  we  have 

(+3)x(-4)  =  (-4)x(4-3). 

But  by  §  17,  (-  4)  X  (H-  3)  =  -  12  =  -  (3  X  4). 

Whence,         (-f  3)  x  (-  4)  =  -  (3  x  4).  (§9,  4) 

We  then  have  the  following  definition : 

To  imdtiply  a  number  by  a  negative  number  is  to  multiply 
it  by  the  absolute  value  (§  13)  of  the  multiplier^  and  change 
the  sign  of  the  result. 

Thus,  to  multiply  +  4  by  —  3,  we  multiply  +  4  by  -f  3, 
giving  4- 12,  and  change  the  sign  of  the  result. 

That  is,  (4-4)  x  (-3)  =  -12. 

Again,  to  multiply  —  4  by  —  3,  we  multiply  —  4  by 
4-  3,  giving  —  12  (§  17),  and  change  the  sign  of  the  result. 

That  is,  (-  4)  X  (-  3)  =  -f  12.     ; 

19.  From  §§  17  and  18  we  derive  the  following  rule : 

To  multiply  one  number  by  another,  multiply  together  their 
absolute  vcdues. 

Make  the  product  plus  ichen  the  multiplicand  and  multiplier 
are  of  like  sign,  and  minus  when  they  are  of  unlike  sign. 


14  ALGEBBA. 

1.  Multiply  +  8  by  -  5. 

Bytherule,  (+ 8)  x  (- 5)  =  -(8  x  5)  =  - 40,  Ans. 

2.  Multiply  -  7  by  -  9. 

Bytherule,         (- 7)  x  (- 9)  =  +  (7  x  9)  =  + 63,  Ans. 

EXAMPLES. 

Find  the  values  of  the  following : 

3.  (+6)x(-3).  10.  (-24)x(-18). 

4.  (-10)x(+5).  U.  f+4X      /_5 

5.  (-7)x(-6).  I     Tj     I     9 

12.  f-^^>,f-± 


6.  (-12)  X  (+4).  V     ISy     ^     14 

7.  (-8)x(-8).  13.-(-|)x(+|). 

8.  (_15)x(+9).  .       14.    (+,9|)^(_2|). 

9.  (+ll)x(-16).  15.   (-I«)x(-1A). 


ADDITION.  15 


III.   ADDITION  AND  SUBTRACTION  OF 
ALGEBRAIC  EXPRESSIONS. 

DEFINITIONS. 

20.  A  Monomial,  or  Term,  is  an  expression  (§  8)  whose 
parts  are  not  separated  by  the  signs  -f  or  —  ;  as  2 a^,  —Sab, 
or  5, 

2a^,  —  3  ah,  and  +  5  are  called  the  tei'7ns  of  the  expression 
2  X-  —  3  ab  -\-  5. 

A  Positive  Term  is  one  preceded  by  a  -f  sign ;  as  +  5  a. 
If  no  sign  is  expressed,  the  term  is  understood  to  be  posi- 
tive. 

A  Negative  Term  is  one  preceded  by  a  —  sign ;  as  —Sab. 
The  —  sign  can  never  be  omitted  before  a  negative  term. 

21.  If  two  or  more  numbers  are  multiplied  together,  each 
of  them,  or  the  product  of  any  number  of  them,  is  called  a 
Factor  of  the  product. 

Thus,  a,  b,  c,  ab,  ac,  and  be  are  factors  of  the  product  abc. 

22.  Any  factor  of  a  product  is  called  the  Coefficient  of 
the  product  of  the  remaining  factors. 

Thus,  in  2 ab,  2  is  the  coefficient  of  ab,  2 a  of  b,  a  ot  2b,  etc. 

23.  If  one  factor  of  a  product  is  expressed  in  numerals, 
and  the  other  in  letters,  the  former  is  called  the  numerical 
coefficient  of  the  latter. 

Thus,  in  2  db,  2  is  the  numerical  coefficient  of  ah. 

If  no  numerical  coefficient  is  expressed,  the  coefficient  1 
is  understood ;  thus,  a  is  the  same  as  1  a. 


16  ALGEBRA. 

24.  By  §  19,  (-  3)  X  a  =  -  (3  X  a)  =  -  3a. 

That  is,  —3a  is  tlie  product  of  —  3  and  a. 

Then,  —  3  is  the  numerical  coefficient  of  a  in  —3  a. 

Thus,  in  a  negative  term,  the  numerical  coefficient  includes 
the  sign. 

25.  Similar  or  Like  Terms  are  those  which  do  not  differ 
at  all,  or  else  differ  only  in  their  numerical  coefficients ;  as 
2  sc^y  and  —  7  x^y. 

Dissimilar  or  Unlike  Terms  are  those  which  are  not  simi- 
lar ;  as  3  x^y  and  3  xy^. 

ADDITION  OF  MONOMIALS. 

26.  The  sum  of  a  and  &  is  a  -f  6  (§  3) ;  and  the  sum  of  a 
and  —b  is  expressed  a  +  (—  6). 

27.  Required  the  sum  of  a  and  —  b. 

By  §  10,  if  a  man  incurs  a  debt  of  $  4,  we  may  regard 
the  transaction  either  as  adding  —  $  4  to  his  property,  or  as 
subtracting  $  4  from  it. 

That  is,  adding  a  negative  number  is  equivalent  to  subtract- 
ing a  positive  number  of  the  same  absolute  value  (§  13). 

Thus,  the  sum  of  a  and  —  6  is  obtained  by  subtracting 
6  from  a. 

Or,  a +  (-&)=  a -6. 

28.  It  follows  from  §§  26  and  27  that  the  addition  of 
monomials  is  effected  by  uniting  them  with  their  respective 
signs. 

Thus,  the  sum  of  a,  —b,c,—  d,  and  —  e  is 

a  —  b-i-c  —  d  —  e. 

It  is  immaterial  in  what  order  the  terms  are  united,  pro- 
vided each  has  its  proper  sign. 


ADDITION.  17 

Hence,  the  above  result  may  also  be  expressed 
c  +  a  —  e  —  d  —  b, 
—  d  —  b-\-c  —  e  +  a,  etc. 

29.  If  the  same  number  be  both  added  to,  and  sub- 
tracted from  another,  the  value  of  the  latter  will  not  be 
changed  (§  9). 

That  is,  a  +  b  —  b  =  a. 

Hence,  terms  of  equal  absolute  value,  but  opposite  sign, 
in  an  expression,  neutralize  each  other,  or  cancel 

30.  To  multiply  4  by  5  +  3,  we  multiply  4  by  5,  and  then 
4  by  3,  and  add  the  second  result  to  the  first. 

In  like  manner,  to  multiply  a  by  6  +  c,  we  multiply  a  by 
b,  and  then  a  by  c,  and  add  the  second  result  to  the  first. 

That  is,  a(b  -h  c)  =  ab  -\-  ac. 

31.  Addition  of  Similar  Terms  (§  25). 

1.  Find  the  sum  of  5  a  and  3  a. 

We  have,  6 a  +  3  a  =  (5  +  3)a  (§30) 

=  8  a,  Ans. 

2.  Find  the  sum  of  —  5  a  and  —  3  a. 

We  have,      (- 5a)  +  (- 3a)  =  (- 6)  x  a +  (- 3)  x  a  (§19) 

=  [(-5)  +  (-3)]xa  (§30) 

=  (-8)xa  (§15) 

= -8a,  Ans.  (§19) 

3.  Find  the  sum  of  5  a  and  —  3  a. 

We  have,  6a +(- 3)a  =[5 +(- 3)]  x  a  (§30) 

=  2  a,  Ans.  (§16) 


18  ALGEBRA. 

4.   Find  the  sum  of  —  5  a  and  3  a. 

We  have,  (_  5)a  +  3a  =[(- 5)+ 3]  x  a  (§30) 

=  (-2)x«  (§15) 

=  —  2a,  Aus. 

Therefore,  to  add  ttvo  similm'  terms,  find,  the  sum  of  their 
numerical  coefficients  (§§  15,  24),  and  affix  to  the  result  the 
commoyi  letters. 

EXAMPLES. 

Add  the  following : 

5.  5a  and  —12a.  9.  —be  and  6bc. 

6.  —  7  m  and  —  8  m.  10.  xyz  and  —  9  xyz. 

7.  15  X  and  -llx.  11.  -18??iV  and  -27mV. 

8.  -lOa^andlal  12.  86  a^ftc^  and  -  19  a^ftc^. 

13.  Eequired  the  sum  of  2  a,  —  a,  3  a,  —  12  a,  and  6  a. 

Since  the  order  of  the  terms  is  immaterial  (§  28),  we  may  add  the 
positive  terms  first,  and  then  the  negative  terms,  and  finally  combine 
these  two  results. 

The  sum  of  2  a,  3  a,  and  6  a  is  11  a. 

The  sum  of  —  a  and  —  12  a  is  —  13  a. 

Then  the  required  sum  is  11  a  +  (—  13  a),  or  —  2a,  Ans. 

Add  the  following : 

14.  9  a,  —  7  a,  and  8  a.         15.   13  x,  —  x,  —  10  Xy  and  5  x. 

16.  12  abc,  abc,  —  6  abc,  and  —  17  abc. 

17.  15  m^,  —  11  m^,  —  4  m^,  m^,  and  14  ml 

18.  21  a^2/^  -  16  x^,  -  x^y\  3  x^y\  and  -  19  3^y\ 

If  the  terms  are  not  all  similar,  we  may  combine  the 
similar  terms,  and  unite  the  others  with  their  respective 
signs  (§  28). 


ADDITION.  19 

19.  Kequired  the  sum  of  12  a,  —5  a;,  —Sy^,  —5  a,  8  a;, 
and  —  3  x. 

The  sum  of  12  a  and  —  5  a  is  7  a. 

The  sum  of  -  5 x,  8  a;,  and  -  3  a;  is  0  (§  29). 

Then  the  required  sum  is  7  a  —  3  y'^,  Ans. 

Add  the  following  : 

20.  Sab,  —7 cd,  —  5 a6,  and  3 cd. 

21.  Qx,  —lOz,  2  y,  4  2,  —  9  y,  and  —  x. 

22.  12"m%  -  2  wi,  -  8  n,  5,  -  3  n,  -  7  m^,  and  11  n. 

23.  10a,  -Gd,  -5c,  126,  -  a,  c,  -3c,  and  -9a. 

24.  7  X,  —4:y,  —  Sz,  9y,  —  2x,  —  Sx,  —  5  2;,  6  ?/,  and  —  z. 

DEFINITIONS. 

32.  A  Polynomial  is  an  algebraic  expression  consisting  of 
more  than  one  term ;  as  a  -f  6,  or  2  a;^  —  3  a;y  —  5  y'i 

A  Binomial  is  a  polynomial  of  two  terms ;  as  a  -h  b. 
A  Trinomial  is  a  polynomial  of  three  terms. 

33.  A  polynomial  is  said  to  be  arranged  according  to  the 
descending  powers  of  any  letter,  when  the  term  containing 
the  highest  power  of  that  letter  is  placed  first,  that  having 
the  next  lower  immediately  after,  and  so  on.     Thus, 

x^-\-Safy-2a^y"-{-3xy  -4:y* 

is  arranged  according  to  the  descending  powers  of  x. 

Note.  The  term  —  4  y^,  which  does  not  involve  x  at  all,  is  regarded 
as  containing  the  lowest  power  of  x  in  the  above  expression. 

A  polynomial  is  said  to  be  arranged  according  to  the 
ascending  powers  of  any  letter,  when  the  term  containing 
the  lowest  power  of  that  letter  is  placed  first,  that  having 
the  next  higher  immediately  after,  and  so  on.     Thus, 

x''-\-3^y-2xY  +  3^-^y^ 
is  arranged  according  to  the  ascending  powers  of  y. 


20  ALGEBRA. 

ADDITION  OF  POLYNOMIALS. 

34.  A  polynomial  may  be  regarded  as  the  sum  of  its 
separate  monomial  terms  (§  28). 

Thus,  2a  —  3&  +  4cis  the  sum  of  2 a,  —3b,  and  4 c. 

Hence,  the  addition  of  polynomials  may  he  effected  by  uniting 
their  terms  ivith  their  respective  signs. 

1.  Required  the  sum  of  Ga  —  7oc^,  3a^  —  2a-\-3y^,  and 
2x^  —  a  —  mn. 

It  is  convenient  in  practice  to  set  the  expressions  down  one  under- 
neath the  other,  similar  terms  being  in  the  same  vertical  column. 

We  then  add  the  terms  in  each  column,  and  unite  the  results  with 
their  respective  signs.     Thus, 

6  a  -  7  x2 

—    a  +  2  a;2  —  mn 


3  a  —  2  a;2  +  3  2/3  —  mn,  Ans. 


EXAMPLES. 

Add  the  following : 

2.                              3.  4. 

7a_55             _    8m2+   6n^  -19a6-    led 

-  9  a  +  2  &                 12  m^  -  16  n«  Sab -lied 

3a-    b             -    Gm^ 4-14^3  6ab-\-13cd 


5.  4a-66H-3cand5a  +  26-9c. 

6.  m^  +  2  mw  -f  n^,  m^  —  2mn-\-  n^,  and  2m^  —  2n\ 

7.  3x  —  Sy,ly  —  6z,  and  5z  —  2x. 

8.  2a^-^ab-b%  la''+3ab-^b\  and  -  4a2_6a6 -f- 8  61 

9.  4«-3a;2-ll  +  5ar^,  12ar^-7-8ar«-15a;, 

and  14  +  6a^  +  10a;-9cc^. 


ADDITION.  21 

Note.  It  is  convenient  to  arrange  the  first  expression  in  descending 
powers  of  x  (§  33),  as  follows  : 

5a;3_3^2_,.4a,_n. 

and  then  write  the  other  expressions  underneath  the  first,  similar 
terms  being  in  the  same  vertical  column. 

10.  2a-3b-5c,  Sb -\-6c-{-7  d,   -4a-3c4-2d, 

and  7 a  —  b  —  9d. 

11.  x'-3xy'-2a^y,  3x^y  -  5if  -  Axf,  5xy'' -  6  f -7  j^, 

and  Sf-j-7x^-dxry. 

12.  6a-Sb-2c,  12c-\-9d-7a,llb-10c-5d, 

and  —3b  —  4:d-\-a. 

13.  15a'-2-9a'-3a,    13a  -  5a^ -6 -7  a^, 

8-f  4a-8a3-7a2,  and  IGa^  +  Sa^- 10a  -  2. 

14.  9a2-1362-18c2,    lo  c' -^  12  b' -  S  d', 

19d^-Ua'-\-3c%  and  -  2^^- 16^^  +  lla^. 

15.  12a^-c^-{-4:ax'-5a\  ISa^ -2a'x-3a^ -13ax', 

15a'x  -  lla^  -  17  a^  -\-  3as^, 
and  Gax^-Sd-x-7x^-i-9  a?. 

16.  13.T2+3-4.r4-8a^,   -  9a;  +  5  4- 16a^  H- a^, 

-15-6i«2-7a;3  +  lla', 

and  -  lOar^  -  12a;  +  14a;2  _  17. 

SUBTRACTION. 

35.  Subtraction,  in  Algebra,  is  the  process  of  finding  one 
of  two  numbers,  when  their  sum  and  the  other  number  are 
given. 

The  Minuend  is  the  sum  of  the  numbers. 

The  Subtrahend  is  the  given  number. 

The  Remainder  is  the  required  number. 


22  ALGEBRA. 

36.  The  remainder  when  b  is  subtracted  from  a  is  ex- 
pressed a  —  6  (§  3) ;  and  the  remainder  when  —  b  is  sub- 
tracted from  a  is  expressed  a  —  (—  6). 

37.  Let  it  be  required  to  subtract  —  b  from  a. 

By  §  35,  the  sum  of  the  remainder  and  the  subtrahend 
is  equal  to  the  minuend. 

Therefore,  the  required  remainder  must  be  such  an  ex- 
pression tliat,  when  it  is  added  to  —  b,  the  result  shall 
equal  a. 

Now  if  a  +  6  be  added  to  —  b,  the  result  is  a. 

Hence,  the  required  remainder  is  a  +  b. 

That  is,  a  —  (—  b)  =  a  -\-  b. 

38.  From  §§36  and  37,  we  have  the  following  rule: 

To  subtract  one  number  from  another,  change  the  sign  of 
the  subtrahend,  and  add  the  result  to  the  minuend. 

SUBTRACTION   OF  MONOMIALS. 

39.  1.    Subtract  5  a  from  2  a. 

It  is  convenient  to  place  the  subtrahend  under  the  minuend. 
We  then  change  the  sign  of  the  subtrahend,  giving  —  5  a,  and  add 
the  result  to  the  minuend.    Thus, 

2a 

—  ba 


—  3  «,  Ans. 

2.    Subtract  —5a  from  —2a. 

The  student  should  perform  mentally  the  operation  of  changing  the 
sign  of  the  subtrahend ;  thus,  in  Ex.  2,  we  mentally  change  -  6  a  to 
5  a,  and  then  add  5  a  to  —2  a. 

-2a 

—  6a 


3  a,  Ans. 


9  from  - 

-25. 

9. 

-5  from  16. 

5  from  5. 

10. 

12  from  -17. 

26  from 

-18. 

11. 

-14  from  13. 

14. 

15. 

16. 

-lab 

14  m^n 

21xyz 

11  ah 

-Sm^w 

Ma^z 

SUBTRACTION.  23 

EXAMPLES. 
Subtract  the  following : 

3.  7  from  4.  6. 

4.  4  from  -  11.        7. 

5.  -15  from  -9.     8. 

12.  13. 

15a  -12r^ 

6a  -Six" 

17.  —  xy  from  xy.  21.  —  45aaJ*from  —  19aa^. 

18.  -16a«  from   -Ua'.  22.  SI  a'b^  from  Sa^ft'*. 

19.  21  m'7i^  from  39  mV.  23.  From  8  a  take  -  12  b. 

20.  19  abc  from   -  6  a6c.  24.  From  -  3  m^  take  4  n^ 

25.  From  —23  a  take  the  sum  of  19  a  and  —5  a. 

Note.  A  convenient  way  of  performing  examples  like  the  above 
is  to  write  the  given  expressions  in  a  vertical  column,  change  the  sign 
of  each  expression  which  is  to  be  subtracted,  and  then  add  the  results. 

26.  From  the  sum  of  —  18  xy  and  11  ooy,  take  the  sum 
of  —  29  an/  and  17  ocy. 

27.  From  the  sum  of  26  a^  and  —  7  a^,  take  the  sum  of 
-15a2  and  48a2. 

28.  From  the  sum  of  33  n^x  and  — 16  n%  take  the  sum 
of  49  n\  —  27  n\  and  —  39  n^x. 

SUBTRACTION  OF  POLYNOMIALS. 

40.  A  polynomial  may  be  regarded  as  the  sum  of  its  sep- 
arate monomial  terms  (§  28) ;  hence, 

To  subtract  one  polynomial  from  another,  change  the  sign 
of  each  term  of  the  subtrahend,  and  add  the  result  to  the 
minuend. 


24  ALGEBRA. 

1.    Subtract  7ah^-9  arh  +  8  6^  from  5  a^  -  2  a^^b  +  4  ab\ 

It  is  convenient  to  place  the  subtraliend  under  the  minuend  so  that 
similar  terms  shall  be  in  the  same  vertical  column. 

We  then  mentally  change  the  sign  of  each  term  of  the  subtrahend, 
and  add  the  result  to  the  minuend.     Thus, 

5  a^  -  2  a^h  +  4  ah"^ 

-  9  a2&  +  7  alfi  +  8  ft^ 


5  a3  +  7  a%  -  3  aft^  _  8  6^,  Ans. 


EXAMPLES. 
Subtract  the  following : 

2.       12a2-9a-7  3.       2a64-    56c-3c« 

8  ^2  _  (5  ^  _l_  13  —  a6  +  11  6c  —  4  ca 

4.  From  x-  —  2xy  -{-y^  subtract  o?  -\-2xy  -\-  y^. 

5.  From  5a  —  36  +  4c  subtract  5 a  +  3 6  —  4 c. 

6.  From  4aj3-9a;2+llaj-18  take  3a^-8a^+17a;-25. 

7.  Fi'om  8a7  —  3?/  —  42;  take  —  2;  +  11  a^  —  6 ^. 

8.  Take  76-9c-2cZ  from  6  a  -  5  6  +  12  c. 

9.  Take  12  a^  +  4  a  -  9  from  3  a^  +  8  a^  -  6. 

10.  Subtract  a^- 7  -  2a; -Ga.-^  from  5a;2-12  H-9a^-2a;. 

(See  Note  to  Ex.  9,  page  21.) 

11.  Subtract  1  +  a^  —  a  —  a^  from  3  a  —  3  a^  +  1  —  a^. 

12.  Take  81  5^  +  4  a^  -  36  ah  from  -  30  a6  +  9  a^  +  25  h^. 

13.  From  lOa^  -  21a^  -  11a;  take   -  15 a;^  _  20 a;  +  12. 

14.  From  17  a»  -  12  ah''  +  5  6^  take  8  a=^  -  3  a-6  +  13  h\ 

15.  Take   -o^^Z^y-'^xy^^f  from  x'-^x^y-lxf^-f. 

16.  Take  6c-5(^-96-4a  from  -106-2c+3a-9cZ. 

17.  Subtract  4-3a;-a;2  +  8a;3_^lQ^4 

from  9-7a;  +  6a;2_i2a;3^5^4 


SUBTRACTION.  25 

18.  Subtract  20:^  -  xy -\-Si/ -9x-  Uy 

from  3x^-5xy-\-2y'^-2x  +  7y. 

19.  From  7  a  -  11  a"^  -  8  -f  6  a* 

subtract  16  a^  _  9  +  2  a^  +  lo  a  -  10  a\ 

20.  From  a^  +  Sx^y  -xY'  -\-oa^if-4tX7/*' 

subtract  Sx^y-7  xY  -  Q  3?f -\- 11  xy^  -  f. 

21.  From  d^-^2ab-^b'^  subtract  the  sum  oi  —a^^2ab-b^ 

and  -2d'  +  2b\ 

Note.  Write  the  expressions  one  underneath  the  other,  similar 
terms  being  in  the  same  vertical  column,  change  the  signs  of  the  terms 
of  each  expression  which  is  to  be  subtracted,  and  add  the  results. 

22.  From  the  sum  oi  So?-\-2ab-  b^  and  0  a^-  8  a6  -f  6  6^ 

take  6a--oab-\-5b\ 

23.  Subtract  the  sum  of  9x^  —  Sx-\-x^  and   5—Jt^-{-x 

from  6x^  —  7  X  —  4i. 

24.  Subtract  the  sum  of  x-\-  y  —  Sz  and  —  4:X-\-9y  from 

the  sum  of  dx  —  2y  —  z  and  —  5x  -^  6y  —  7 z. 

25.  Take  the  sum  of  6  —  4  x-^  —  a;  and  5  ic  —  1  —  2  ic^  from 

the  sum  of  2x^ -\-7  —  4:X  —  5x^ 
and  3a^-63^-2-^Sx. 

26.  From  the  sum  of  2a -36  +  4(i  and  2  6  + 4c-3d,  take 

the  sum  of  — 4a  —  46  +  3c  —  2d  and  3 a  —  2 c. 

27.  From  the  sum  of  9  a^  —  a^  —  5  and   3  a^  —  a  +  1,  take 

the  sum  of  -8a3+13a  +  3  and  5a''+2a2-6a. 


26  ALGEBRA. 

IV.    PARENTHESES. 

REMOVAL  OF  PARENTHESES. 

41.  The  expressions    a  —  b-\-(c  —  d) 
and  a  —  b  —  (c  —  d) 

indicate  that  the  expression   c  —  d  is   to   be   respectively- 
added  to,  and  subtracted  from,  a  —  b. 

If  the  operations  be  performed,  we  have  by  §  §  33  and  40, 

a  —  b  +  {c  —  d)  =  a  —  b-{-c  —  d, 
and  a  —  b  —  (c  —  d)  =  a  —  b  —  c-]-d. 

In  the  first  case,  the  signs  of  the  terms  within  the  paren- 
thesis are  not  changed  when  the  parenthesis  is  removed; 
while  in  the  second  case,  the  sign  of  each  term  within  is 
changed,  from  +  to  — ,  or  from  —  to  +. 

We  then  have  the  following  rules : 

A  parenthesis  preceded  by  a  -\-  sign  may  be  removed  without 
changing  the  sigyis  of  the  terms  enclosed. 

A  parenthesis  preceded  by  a  —  sign  may  be  removed  if  the 
sign  of  each  term  enclosed  be  changed,  from  -\-  to  —,  'or  from 
—  to  +. 

42.  The  above  rules  apply  equally  to  the  removal  of  the 
brackets,  braces,  or  vinculum  (§  7). 

It  should  be  noticed  in  the  case  of  the  latter  that  the 
sign  apparently  prefixed  to  the  first  term  underneath  is 
in  reality  prefixed  to  the  vinculum ;  thus,  -{-a  —  b  means 
the  same  as  +  (a  —  b),  and  —a—b  the  same  as  —  (a  —  b). 

43.  1.   Remove  the  parentheses  from 

2a -36 -(5a -46)  + (4a -6). 
By  the  rules  of  §  41,  the  expression  becomes 

2a  —  36  —  5a  +  46+4a  —  6  =  a,  Ans. 


PARENTHESES.  27 

Parentheses  are  often  found  enclosing  others;  in  this 
case  they  may  be  removed  in  succession  by  the  rules  of 
§  41 ;  and  it  is  better  to  remove  first  the  innermost  pair. 


2.    Simplify  4a;-J3a;4-(-2x-a;-a)|. 
Removing  the  vinculum  first,  and  the  others  in  succession,  we  have 


Ax  - {^x  -\-  {-  2x  -  X  -  a)} 
=  4x  -{Sx+(-2x-x-{-  a^} 
=  ix—{Sx  —  2x  —  x-{-a} 
=  ix  —  3x  +2x-\-  X  —  a  =  Ax  —  a,  Ans. 


EXAMPLES. 

Simplify   the    following    expressions    by   removing   the 
parentheses,  etc.,  and  uniting  similar  terms: 


3.  8a  +  (56-a)-(-76  +  2a). 

4.  4m-[2mH-9n] -S-5>?i-6Vij. 


5.  x-i-y  —  z-\-y  —  z  —  X  —  z  —  x-\-y. 

6.  ab  -  4.b^  -  {2a'  -  b^  -\-  5a^  -\-2  ab  -Sb^ 


7.  m^  —  3  mn  -\-5m'  —  mn  —  6  w^  —  [8  m^  —  4  mn  —  7  n^]. 

8.  4a; -(5a;- [3a; -1]). 


9.   a-(6  — c  +  cZ  +  e). 
10.   5 ab  -  [(3 ab  -  10) -(-4:ab- 7)]. 


11.  7ar^  |-(-3ar^H-2a;-5)-(4ic2-6a;-2). 

12.  m-(6m-7w)  -j-3m  +  4:W-(2m-3w)j. 

13.  17  -  [45  -  (9  -  23  -  32)]. 

14.  3(i-(5a-J-7a  +  [9a-4]|). 

15.  a;  -  [2  a;  -  (-  a;  -f  1)  +  3]  -  J6a;  -  [-  (a;  -  3)  -  a;]  J. 

16.  X  -  (y  -^  z  -  [x  ~  {-  X  -  y)  -{-  zj)  -\-lz  -  2x-y\. 


17.   27i-[37i-J4n-n-4|-(-5n-9)]. 


18.   28-j-16-(-4+[55-31-f47])S. 


28  ALGEBRA. 


19.   a-(2a-lSa-\4.a-5a-l\J). 


20.  c-[2c-(6a-6)-Jc-5a+2  6-(-5a+6a~3  6)S]. 

21.  x—[y  —  \x  —  z  —  x  —  y-{-z\-\-{2x  —  \  —  x-\-  y\)]. 

22.  5x-l2x-(-x-{2x-^r^\-Sx)-Sx']. 


23.    a-\-a-[-a-(-a-\-a-a-l\)]\. 

INTRODUCTION   OF   PARENTHESES. 

44.  To  enclose  any  number  of  terms  in  a  parenthesis,  we 
take  tlie  converse  of  the  rules  of  §  41 : 

Any  number  of  terms  may  be  enclosed  in  a  parenthesis 
preceded  by  a  -\-  sign,  without  chayiging  their  signs. 

Any  number  of  terms  may  be  enclosed  in  a  parenthesis  pre- 
ceded by  a  —  sign,  if  the  sign  of  each  term  be  changed,  from 
+  to  —,  or  from  —  to  +. 

1.  Enclose  the  last  three  terms  of  a  —  b-\-c  —  d-\-e  in 
a  parenthesis  preceded  by  a  —  sign. 

Result,  «— 6— (— c  +  d  —  e). 

EXAMPLES. 

In  each  of  the  following  expressions,  enclose  the  last 
three  terms  in  a  parenthesis  preceded  by  a  —  sign: 

2.  a-[-b-c-d.  6.  o? -{-f +  z^ -3xyz. 

3.  a;3-5a^-8a;  +  7.  7.  a-b-c  +  d-^e. 

4.  m^  +  m^n  +  mn^ -f  7i3.  8.  a*  +  Ba^ -f- a^- 9a -f- 2. 

5.  a^  —  b^  -\-2bc—  cl  9.  ^  —  m^  —  2  mn  —  n^. 

10.  In  each  of  the  above  results,  enclose  the  last  two 
terms  in  parenthesis  in  brackets  preceded  by  a  —  sign. 


MULTIPLICATION.  29 


V.  MULTIPLICATION. 

45.  The  Law  of  Signs. 

If  a  and  b  are  any  two  numbers,  we  have  by  §  19, 

(4-  a)  xi+b)  =  -{-  ah,  (-f-  a)  x(-b)  =  -  ab, 

(— a)  X  (-\- b)  =  —  ab,  (— a)  x  (— b)  = -\- ab. 

From  these  results,  we  may  state  the  Rule  of  Signs  in 
Multiplication  as  follows : 

-f-  multiplied  by  +,  and  —  midtiplied  by  — ,  prodxice  +  ; 
+  multiplied  by  — ,  and  —  multiplied  by  +,  produce  — . 

Or,  as  it  is  usually  expressed  with  regard  to  the  product 
of  two  terras. 

Like  signs  produce  -{-,  and  unlike  signs  produce  — . 

46.  The  Index  Law. 

Let  it  be  required  to  multiply  a^  by  a\ 

By  §  6,  a^  =  a  X  a  X  a, 

and  a^  =  a  X  a. 

Whence,  a^xar  =  axaxaxaxa=a^. 

Therefore,  the  exponent  of  a  letter  in  the  jyroduct  is  equal 
to  its  exponent  in  the  multiplicand  plus  its  exponent  in  the  i 
multiplier. 

Or  in  general,  if  ni  and  n  are  any  two  positive  integers, 

a*"  X  a"  =  «'"+". 

A  similar  result  holds  for  the  product  of  three  or  more 
powers  of  a. 

Thus,  a^xa'x  a'  =  a^^'+'  =  a''. 


30  ALGEBRA. 

MULTIPLICATION  OF  MONOMIALS. 

47.  Let  it  be  required  to  multiply  7  a  by  —2  b. 

We  have,  -2h  =  {-2)xh.  (§45) 

Whence,       Tax  (-2&)  =  7ax  (-2)  x  h. 

Then  since  the  order  of  the  factors  is  immaterial  (§  18), 

Tax  (-2Z>)  =  7x(-2)  xax5 

=  -14a6.  (§  19) 

48.  From  §§  45,  46,  and  4T,  we  derive  the  following  rule 
for  the  multiplication  of  two  monomials  : 

To  the  product  of  the.  absolute  values  of  the  yiumerical  coeffi- 
cients, annex  the  letters;  giving  to  each  an  exponent  equal  to  its 
exponent  in  the  multiplicand  plus  its  exponent  in  the  mtdtiplier. 

Make  the  product  -f-  when  the  midtiplicand  and  multiplier 
are  of  like  sign,  arid  —  when  they  are  of  unlike  sign. 

1.  Multiply  2  a^  by  9  al 

By  the  rule,  2a5x9a4  =  2x9x  a^+^  =  18  a?,  Ans. 

2.  Multiply  a^Wc  by  -^w'bd. 

We  have,        a^&'^c  x  (-  5  a%d)  =  -  5  a^b^cd,  Ans. 

3.  Multiply  -  Taj"*  by  4^3 

We  have,  (  -  7  x'»)  x  4  x^  =  -28  x'"+%  Ans. 

4.  Multiply  -3ic"by  -8a;^ 

We  have,      (  -  3  a:")  x  (  -  8  x«)  =  24  a;"+»  =  24  x^»,  Ans. 

EXAMPLES. 
Multiply  t!ie  following : 

5.  7a^  by  3a^  7.   5xyz  by  —llxyz. 
?,    -}4a6b^2cd  8.    -  15  a%  hj  -  4:  ah' , 


MULTIPLICATION.  31 

9.  -9mVby7mV.  13.  -  a'»6"c^  by  -  a6V. 

10.  -6a^b^  by  -  6V.  14.  -Sary*"  by  Ux'^y^ 

11.  Sa^z'  by  -8/2^.  15.  10a*6V  by  9a'<f(P. 

12.  12  a^ftc  by  ebcd\  16.  IGaj^Pi/'  by  -Sa^^?/'". 

49.  We  have  by  §  45, 

(— a)x(—  &)x(—  c)  =  (a6)  x(—  c)=—abc;         (1) 
(-  a)  X  (-  6)  X  (  -  c)  X  (-  d)  =  (-  abc)  x  (-  d),  by  (1), 

=  abed ;  etc. 

That  is,  the  product  of  three  negative  terms  is  negative ; 
the  product  of  four  negative  terms  is  positive ;  and  so  on. 

Hence,  the  product  of  any  number  of  terms  is  positive  or 
negative  according  as  the  number  of  negative  terms  is  even  or 
odd. 

1.   Required  the  product  of  —2a^W,  6  bc^,  and  —  7  (?d. 
Since  there  are  two  negative  terms,  the  product  is  positive. 
Whence,      ( -  2  a%^)  x  (6  h&)  x{-lcH)  =  84  a^h^c'd,  Ans. 

EXAMPLES. 
Multiply  the  following : 

2.  Sa"*,  5a«,  and  -6a*. 

3.  -A.x\  -9?/,  and  2z\ 

4.  .'»^'"2/",  y'^zF,  and  ar'z'. 

6.    -12a'b\  -b'(?,  and   ~^&a\ 

6.   a\  3  a,  5  a\  and  —  7  a^. 

7.-2  a^ftSc^  2  a^bd\  -  2  ac^d^  and  2  6Vd. 

8.  2a^^S  --32/^^  -  ^^x\  and  -ba^y'^T^. 

9.  —  a'^af ,  —  ft^"?/,  —  a^'"2/',  and  —  6"a;. 

JQ.   5a6^  -4aV,  -a«d3,66%  and  -Sc^d^ 


32  ALGEBRA. 

MULTIPLICATION  OF  POLYNOMIALS  BY  MONOMIALS. 

50.  In  §  30,  we  showed  that  the  product  of  a  +  b  and  c 
was  ac  +  he. 

We  then  have  the  following  rule  for  the  product  of  a 
polynomial  by  a  monomial : 

Multiply  each  term  of  the  polynomial  by  the  monomial,  and 
unite  the  results  ivith  their  proper  signs. 

1.    Multiply  -8  0^  by  2^2- 5  ic-f  7. 
Multiplying  each  term  of  2x^  —  bx  +  7  by  —  8 .x^,  we  have 
(2ic2_5x  +  7)x(-8x3)  =  -  16x5  +  40x4-56x3,  Ans. 

EXAMPLES. 

Multiply  the  following  : 

2.  4  a  —  9  by  5  a.  7.   mhi^  by  m^  —  2  mn  -f-  n^. 

3.  Sx'y-Bxfhy   -Sxy\  8.   H oFb'' -  9 ab' hy  -  6 a'b\ 

4.  a^-ab-{-b^  by  ab.  9.   6x^-5x^-7 x' hj  -Sx^. 

5.  3aj2  4-  a:  -  8  by  -  9aj2.  (See  note  to  Ex.  9,  p.  21.) 

6.  -7a^  hj  2a^-6a'-7.      10.    -W-a^-{-5abhj  4:a%\ 

11.  —  x^y^  by  a^  —  3  x^y  -^  3  xy^  —  y^. 

12.  5a'  +  d-Sa'-4.a-a'  hj  7a'. 

13.  —  2  mn  by  3  m^  —  6  mSi  —  7  mn^  4-  2  n^. 

MULTIPLICATION  OF  POLYNOMIALS  BY  POLYNOMIALS. 

51.  Let  it  be  required  to  multiply  a  -\-  b  hy  c  -\-  d. 

As  in  §  30,  we  multiply  a-\-b  by  c,  and  then  a-\-  bhj  d, 
and  add  the  second  result  to  the  first. 

That  is,     (a  +  6)  (c  +  d)  =  (a  +  b)c  +  (a  +  b)d 
=  ac  4-  6c  +  ad  -f  bd. 


MULTIPLICATION.  33 

We  then  have  the  following  rule : 

Multiply  the  rmiltijylicand  by  each  term  of  the  multiplier, 
and  add  the  partial  products. 


52.   1.   Multiply  3a-46  by  2a-5b. 

Ill  accordance  with  the  rule,  we  multiply  3  a  —  4  5  by  2  a,  and 
then  by  —  5  6,  and  add  the  partial  products. 

A  convenient  arrangement  of  the  work  is  shown  below,  similar 
terms  being  in  the  same  vertical  column. 

3a  -46 
2a  -.bb 


6a2-   Sab 

-15a6  +  20  62 

6  a2  -  23  ab  +  20  b^,  Ans. 

Note.  The  work  may  be  verified  by  performing  the  example  with 
the  multiplicand  and  multiplier  interchanged. 

2.   M ultiply  4 ax"  -i-a^ -Sa^ -2a'x  by  2x  +  a. 

It  is  convenient  to  arrange  the  multiplicand  and  multiplier  in  tlie 
same  order  of  powers  of  some  common  letter  (§  33),  and  write  the 
partial  products  in  the  same  order. 

Arranging  the  expressions  according  to  the  descending  powers  of 
a,  we  have 

a'»-2a%  +  4ax2  -Sx» 
a  +2x 

a*-2a^x  +  4  aH^  -  8  ao^ 

2  aH  -  4  aV  +  8  ax^  -  16  x^ 


a*  —  16  X*,  Ans. 

EXAMPLES. 

Multiply  the  following : 

3.  2a  +  5by3a  +  7.  6.    -  7a6  +  2  by  -  4o6- 6. 

4.  5  a  —  8  by  6  a  —  1.  7.    of  —  xy  -{-y-  by  x  +  y. 

6.   —4:X  —  5ybySx-\-3y.     8.   2 a=*  -h  7 a  —  9  by  5 a  —  1. 


34  ALGEBRA. 

9.  Sx'-x  +  4.hj  4:X-3. 

10.  -  8  71  +  5  n2  -  3  by  2  +  w. 

11.  3  a  -  2  6  by  9  a^  +  6  a6  +  4  b\ 

12.  a  —  b-{-chja  —  b-\-c. 

13.  6m^  —  5  mn  —  Sn^  by  2  m*^  +  3  m?n. 

14.  a^  +  3a^-7aj-6  by  3,T-4. 

15.  m^  +  mn  +  71^  by  m^  —  7?i7i  +  n^. 

16.  8a=^-4a2  +  2a-l  by  2a  +  l. 

17.  ^x'-^-^Q^xhy  ^x^^  +  li^. 

18.  67i-8-f-4n2  by-4  +  2n2-3?L 

19.  3a2-5a6-862  by  4a^-9a6-762. 

20.  2ic  +  62-42/  by  22/-30  +  .'c. 

21.  4a  +  66  +  10cby2a-36  +  5c. 

22.  a«-2a2_}-a-2  by  a2  +  2a  +  3. 

.         23.  a;^  +  2a^  +  4ic2_^8a;-hl6  by  i»-2. 

24.  m^  -\-n^  +  mn^  +  m^n  by  m^n  —  mn^. 

25.  -5a^  +  9  +  2aj«-4a;  by  5a^-l4-6a;. 

26.  4  a^*"?/  -  5  a^2/'"~'  by  2  a^-y +*  -  3  a:^/". 

27.  3m3-5m2-}-4m-l  by  2m2-m--3. 

28.  16a^-24a3  +  36a2-54a  +  81  by  2a-{-3. 

29.  a^  -  3  a^ft  +  3  aft^  _  53  ^y  ^2  _  2^6  +  52. 

30.  a;«-2a.-2-a;-l  by  a^ -\-2x' -  x-^1. 

31.  a^  -  6  a2  + 12  a  -  8  by  a^  +  6  a^  +  12  a  +  8. 

32.  m^  —  6  mn  —  In^  by  m^  —  2  m^n  —  5  mn^  -|-  4  n^. 

33.  a;«-34-2a^-aj  by  3-ic+aj3-2aj2 

34.  a?  +  y^  -\-  c^  -{- ah  —  he -[-  ca  by  a  —  &  —  c. 

35.  e>^-4:X^y-3xy^  +  2f  hy  2x'  +  xy-2y\ 


MULTIPLICATION.  35 

Find  the  product  of  the  following : 

36.  x  —  2,  x  —  S,  and  ic  —  4. 

37.  a -\- 5,  2  a  — 3,  and  4  a  —  1. 

38.  x  —  y,  x^  -\-xt/  -{-  y^,  and  x"  -\-if 

39.  3?i  — 8,  4  7t  +  7,  and  hn  —  ^. 

40.  a  —  x^  Oj-\-x,  o?  -\-  x^f  and  a'^  -\-  a^. 

41.  m  —  4  ?j,  2  m  +  3  n,  and  2  7?i^  -f  5  m7i  — 12  7i^ 

42.  a  -f  1,  a  —  1,  a^  +  a  -h  1,  and  a^  —  a  +  1. 

43.  a^  -H  a;  +  1,  a^  —  a;  + 1,  and  a^  —  a:^  +  1. 

44.  a-\-h,  a  —  b,  2a  — 3 b,  and  2a +  36. 

46.   a;  +  3,  2a;-|-l,  2a;-l,  and  4a^- 12ar^-h  x- 3. 

53.   1.    Simplify  (a  -  2a;)2- 2(a;-f  3a)(a  -  a:). 

To  simplify  the  expression,  we  should  first  multiply  a-2x  by 
itself  (§  6) ;  we  should  then  find  the  product  of  2,  x  +  3  a,  and  a  -  x, 
and  subtract  the  second  result  from  the  first. 


a  -2x 
a  -2x 

3a  +x 
a  -X 

a2-2ax 

-  2  ax  +  4  x2 

3a2+    ax 

-Sax-    x2 

a2-4ax  f  4x2 

3a2_2ax-    x2 
2 

6a2-4ax-2x2 

Subtracting  the  second  result  from  the  first,  we  have 

a2  -  4  ax  +  4x2  -  6a2  +  4ax  +  2x2  =  -  5a2  4-  6x2,  AiiS. 

EXAMPLES. 
Simplify  the  following : 

2.  (3a;-8)(a;  +  6)  +  (2a;-7)(4a;4-9). 

3.  (2a  +  6)(3a-7)-(2a-5)(3a  +  7). 


36  ALGEBRA. 

4.  (a  —  m)  (b  +  n)  +  (a  +  m)  (6  —  n). 

5.  (x-y-\-zy-(x  +  y  -  zf. 

6.  (a-b-c  +  ay. 

7.  (2x  +  3y(2x-3y. 

8.  (a  +  &)  (a'  -  6^)  -(«-&)  (a^  +  b'). 

9.  (3i«-52/)'-5(a;-?y)(aj-52/). 

10.  (a  +  x)  (a^  4-  a^'')  [a'  -  .t  (a  -  a;)]. 

11.  (a  -  b)  (a'  +  a'b  +  a5^  +  6^)  [(a^  +  &2)2  _  2^252]. 

12.  (x  -\-l)(x  +  2)  (a;  +  3)  -  (a;  -  1)  (x  -2)(x-  3). 

13.  (x  -y){y-z)-  (x-z) (y-z)-(x- y)  (x  -  z). 

14.  (aJ^b  +  cy-  {a  +  by  -  c(2  a  +  2  6  +  c). 

15.  (a  +  l)(2a  +  5)(4a-3)  +  (a-l)(2a-5)(4a  +  3). 

16.  (aj  +  2/-^)'  +  (2/  +  ^-a')'H- (24-05-2/)'. 

17.  2(a  +  2x)(a-2x)l(a  +  2xy-\-{a-2xy]. 

18.  (a  +  &  +  c)2  -  (a  +  &  -  c)2  _  (a  -  6  +  c)^  +  (ct  -  5  -  c)^ 

19.  [(m  +  ny-\-{7n-7iy]l{2m-\-7iy-(m-2  7iyi 

20.  (a  +  ft  -  c)  (6  +  c  -  a)  (c  +  a  -  6). 

21.  (a-}- by -(a -by. 

22.  (^•  +  2/  +  2)'-3(aj24-2/'  +  2;2)(a;  +  2/  +  4 


DIVISION.  37 


VI.    DIVISION. 

54.  Division,  in  Algebra,  is  the  process  of  finding  one  of 
two  numbers,  when  their  product  and  the  other  number 
are  given. 

The  Dividend  is  the  product  of  the  numbers. 
The  Divisor  is  the  given  number. 
The  Quotient  is  the  required  number. 

55.  The  Law  of  Signs. 

Since  the  dividend  is  the  product  of  the  divisor  and  quo- 
tient, the  equations  of  §  45  may  be  written  as  follows : 

(4-  ah)  -  (4-  a)  =  +  6,  (-  a6)  ^  (-h  a)  =  -  h, 

{— ab)  ^  {- a)  = -\- h,  {-\- ah)  ^  {— a)  =  -  h.      * 

From  these  results,  we  may  state  the  Rule  of  Signs  in 
Division  as  follows : 

-f  divided  by  +,  and  —  divided  hy  — ,  produce  +  ; 

+  divided  hy  —,  and  —  divided  by  -j-,  produce  — . 

Hence,  in  Division  as  in  Multiplication, 

Like  signs  produce  -j-,  and  unlike  signs  produce  ~. 

56.  The  Index  Law. 

Let  it  be  required  to  divide  a*  by  a^ 

The  quotient  must  be  a  number  which,  when  multiplied 
by  the  divisor,  a-,  will  produce  the  dividend,  a^. 
Now  if  d^  be  multiplied  by  a^,  the  product  is  a^. 

Whence,  ^  =  a'. 

a^ 

Hence,  the  exponent  of  a  letter  in  the  quotient  is  equal  to 
its  exponent  in  the  dividend  minus  its  exponent  in  the  divisor. 


38  ALGEBRA. 

DIVISION  OF  MONOMIALS. 

57.  Let  it  be  required  to  divide  —  14  a?h  by  7  a^. 

We  find  a  number  which,  when  multiplied  by  7  a^,  will 
produce  — 14  a?h. 

That  number  is  evidently  —  2  6. 

Whence,  rLl4^  =  _2  6. 

i  Qi 

58.  From  §§  55,  ^Q,  and  57,  we  derive  the  following  rule 
for  the  division  of  two  monomials : 

To  the  quotient  of  the  absolute  values  of  the  nuinerical 
coefficients,  annex  the  letters;  giving  to  each  an  expo7ient 
equal  to  its  exponent  in  the  dividend  minus  its  exponent  iii 
the  divisor,  and  omitting  any  letter  having  the  same  exponent 
in  the  dividend  and  divisor. 

Make  the  quotient  +  when  the  dividend  and  divisor  are  oj 
like  sign,  and  —  when  they  are  of  unlike  sign. 

1.  Divide  54  a^  by  -9a*. 

By  the  rule,  -^^^  =  -  6  a'-*  =  -  6  a*,  Ans. 

—  9  a* 

2.  Divide  -  2  a^b^c^'  by  abd\ 

We  have,  =il^^P^  =  -2a-^bc,  Ans, 

abd^ 

3.  Divide  -  91  x^'^y^'z'  by  -  13  x'^y''^. 

We  have,     ~  ^^  ^^"^^"^  =  7  oc^^r^-m^-^  =  7  x^z^-^  Ans. 
—  13  x^y"z^ 

EXAMPLES. 
Divide  the  following : 

4.  35  by  -5.  6.    -64  by  -4. 

5.  -44  by  11.  7.    -84  by  7. 


DIVISION.  39 

8.  -144  by  -8.  17.   40  mV  by  5mhi. 

9.  168  by  -12.  18.    -33aVy*  by  -3a'y. 

10.  16  a'  by  A  a*.  19.  -36a=^+^  by  12  a^— 3. 

11.  -18  0^2/  by  2a^.  20.  81a*6V  by  9  5V. 

12.  2mV  by  -  m V.  21.  65af'2r^2'  by  -13a^. 

13.  -  a«6V  by  -  a^b^c.  22.  -  a^b'  by  -  a«6*. 

14.  -6xy^  by  6a^/.  23.  540^1^  by  9iB«y". 

15.  -24a*62by  _8a^6l  24.  98aVc«  by   -  14  a^jc^. 

16.  28ar'2/23  by   ^ja^z\  25.     -USmVp^  by  llmVi>3. 

DIVISION  OF  POLYNOMIALS  BY  MONOMLAXS. 

59.  We  have,       a(b-{-  c)  =  ab-{-  ac. 

Since  the  dividend  is  the  product  of  the  divisor  and  quo- 
tient (§  54),  we  may  regard  ab  +  ac  as  the  dividend,  a  as 
the  divisor,  and  6  +  c  as  the  quotient. 

Whence,  ?^±^=b  +  c. 

a 

We  then  have  the  following  rule : 

Divide  each  term  of  the  dividend  by  the  divisor,  and  unite 
the  results  with  their  proper  signs. 

1.   Divide  9  a362_  6  a*c  + 12  a^ftc' by  -Sa\ 

^am-Qa^c+l2a%c^  =  _  Safe^  +  2  a^c  -  4  6c3,  Ans. 
—  oa^ 

EXAMPLES. 
Divide  the  following : 

2.  16a^  + 282^-24.^3  by  4a:«. 

3.  104m7i3-39m3n  by  -  13m7i. 

4.  6a26V-15a«6V  +  3a^6«c  by  -Sa^ft^ 


40  ALGEBRA. 

5.  -63a^/;22_  34^32^4^7  ^^^  7  a^yz\ 

6.  20  m^rv'  —  45  m'^n'  —  35  wi%^  by  —  5  mW. 

7.  -  24  a"  + 108  a-*  -f  84  a'  by  12  a«. 

8.  40  a^hc  -  24  ah\  -  32  aftc^  by  -  8  ahc. 

9.  72aji«  -  9a^  +  54a;«  -  ^9x^  by  -  9a;^ 

10.  -  2x^  +  6a^/  -  Gic'^/^  +  2x1/^  by  -  2a?2/. 

11.  60  a}"^  -  30  a^  +  15  a}^  -  45  o?  by  15  a\ 

12.  a^'"6''  -  3  a^'^ft^'*  +  2a'»6^  by  a'"6". 

13.  48  a'h^d"  +  36  a^h^(^  -  30  a«6V  by  6  a^6^c^ 

14.  -  88  xYz"  +'6b  xfz^  +  66  i^f^  by  -  11  a;?/V. 

15.  i»'*+ V+^2:^  —  icy^;'' —  a?'"!/*^;'"  by  —x^'if^^. 

DIVISION  OF  POLYNOMIALS  BY  POLYNOMIALS. 

60.  Let  it  be  required  to  divide  12  +  10  ar  —  11  a?  —  21  a? 
by  2x-2-4-3a;. 

Arranging  each  expression  according  to  the  descending 
powers  of  x  (§  33),  we  are  to  find  an  expression  which, 
when  multiplied  by  the  divisor,  2  a^  —  3  a?  —  4,  will  produce 
the  dividend,  10  x^  -  21  x""  -  11  a;  +  12. 

It  is  evident  that  the  term  containing  the  highest  power 
of  X  in  the  product  is  the  product  of  the  terms  containing 
the  highest  powers  of  x  in  the  multiplicand  and  multiplier. 

Therefore,  10  a^  is  the  product  of .  2  a;^  and  the  term  con- 
taining the  highest  power  of  x  in  the  quotient. 

Whence,  the  term  containing  the  highest  power  of  x  in 
the  quotient  is  10  m?  divided  by  2  oi?,  or  5  x. 

Multiplying  the  divisor  by  5  a;,  we  have  the  product 
10  a;^  —  15  a^  —  20  a; ;  which,  j^^hen  subtracted  from  the  divi- 
dend, leaves  the  remainder  —  6  a?^  +  9  a?  +  12. 

This  remainder  must  be  the  product  of  the  divisor  by  the 
rest  of  the  quotient ;  therefore,  to  obtain  the  next  term  of 
the  quotient,  we  regard  —  6  x^  +  9  x  +  12  as  a  new  dividend. 


DIVISION.  41 

Dividing   the  term  containing  the  highest  power  of   x, 

—  6  ic^,  by  the  term  containing  the  highest  power  of  x  in 
the  divisor,  2  a^,  we  obtain  —  3  as  the  second  term  of  the 
quotient. 

Multiplying  the   divisor  by  —  3.  we  have   the   product 

—  6  ic^  -f  9  .X  -}-  12 ;  which,  when  subtracted  from  the  second 
dividend,  leaves  no  remainder. 

Hence,  5  a?  —  3  is  the  required  quotient. 

It  is  customary  to  arrange  the  work  as  follows : 


10ar^-21ar^-llaj-M2 
10ar^-15a^-20ic 


Is?  —  '6x  —  4,  Divisor. 


5  a;  —  3,  Quotient. 


60^2+    9a; +  12 
6a;2+    9a; +12 


Note.  The  example  might  have  been  solved  by  arranging  the 
dividend  and  divisor  according  to  the  ascending  powers  of  x. 

From  the  above  example,  we  derive  the  following  rule : 

Arrange  the  dividend  and  divisor  in  the  same  order  of 
powers  of  some  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  remit  as  the  first  term  of  the  quotient. 

M^dtiply  the  ivhole  divisor  by  the  first  term  of  the  quotient, 
and  subtract  the  product  from  the  dividend. 

If  there  be  a  remainder,  regard  it  as  a  neio  dividend,  and 
proceed  as  before;  arranging  the  remainder  in  the  same  order 
of  poivers  as  the  dividend  and  divisor. 

61.   1.   Divide  dab''  -\-  a^  -9 b^  -  5 a^bhj  Sb^  -\-  a^  -2 ab. 

Arranging  according  to  the  descending  powers  of  a, 
«8  -  6  a2&  +  9  a62  -  9  68  I  g"^  -  2  gfe  +  3  &2 
gs  _  2 ggft  4-  3 ab^  \  a-Sb,  Ans. 

-  3  g26  +  6  gftg  -  9  b^ 

Note  1.  In  the  above  example,  the  last  term  of  the  second  divi- 
dend is  omitted,  as  it  is  merely  a  repetition  of  the  term  directly  above. 


42  ALGEBRA. 

Note  2.     The  work  may  be  verified  by  multiplying  the  quotient 
by  the  divisor,  which  should  of  course  give  the  dividend. 

2.   Divide  8 +18 0^^-56.^2  by  -  ex' +  4.-\-8x. 

Arranging  according  to  the  ascending  powers  of  x, 

4  +  8^-6x2)8  -  56a;2  +  18x*(2  -  4x  -  3x-2,  Ans. 
8  +  16  X  -  12  x-^ 

-16x  -44x2  +  18x* 
-16x  -32x2  +  24x3 


-  12x2-24x3+  18x* 

-  12x2-24x3+  18x4 


EXAMPLES. 
Divide  the  following : 
s/     3.   Wx'-llx-U  hj  3x-\-2. 
T  4.    25m2  +  40mn-h  16  7^2  by  5m+4n. 
^  5.   12^2- 28a +  15  by  6a -5. 
<Sl   a^-6x^-19x-\-S4:  by  x-7. 

7.  8m3  +  27n3  by  2m  +  3w.   ' 

8.  a^  — 64?/^  by  cc  — 4?/. 

V9.   8  -  16a  +  6a'  by  3a -2. 
^10.   BOx'y^-lS  by  3-5xy. 
/ll.   10a^62_ig^,254_3^353  by  2a%-3ah\ 

12.   2  m''  —  8  iA  + 18  mn-^  by  2  7«,'  —  6  mn. 
A  13.   20  +  36a«-49a  by  12a2  +  5-16a. 

14.  2  a*62  _  3  ^4^3  _  7  ^3^4  _^  4  ^255  ^y  ^35  _  ^252  _  4  ^2>3. 

15.  a^  -  52  +  2  6c  -  c^  by  a  +  6  -  c. 

16.  4^- 16ic2?/ +  60;/  + 6a^  by  3a;2-/-2aj^. 
^     17.   39mn2  +  30m3-20n3-43m2n  by  6m -5n. 


DIVISION.  43 

18.  4a^-9a2-^30a-2o  by  2a- -^  3a -5. 

^19.  4:X  +  x*-\~3  by  Si-x^-2x. 

20.  n'  -  16  by  2^^  +  8  -\-^7i-{-n^ 

^  21.  6m^-19m^-{-22m-\-5  by  3??i-5. 

22.  a.-*  +  y^  +  a^/  by  y^ -\- x^  —  xy. 

23.  l-16a«  by  lH-2al 

24.  16a;^-8l2/'  by  2x-3y. 

25.  -  9m2  -  16  +  ??i^  -  24 m  by  3  m  +  ?/i2  +  4. 

26.  9a;*+4-13a^  by  3ar'-2  +  a;. 

27.  2a* -a^.^  8a -5  by  2a2-3a  +  5. 

28.  13a^  +  71a;-70a^-20+6a:*  by  4  +  3ir'-7a;. 

29.  4 mV  +  w»  +  16m*  by  2mn2  +  4:m^  +  w*. 

30.  a^  +  32  by  a  +  2.   . 

31.  120a*  +  26a3  -  llla^  -  14a  +  24  by  (3a  +  2)(4a  -  3). 

32.  (2  m^  -  m  -1)(3 m^  -h  m  -  2)  by  (2m  +  1)(3 m  -  2). 

33.  a^  +  243  by  9a2  +  81-3a3-27a4-a*. 

34.  4ar^'«+y-16af+y+^  +  12ar'i/^"-^  by  af^+^y-3xy''-\ 

35.  6a'-6ab'  by  -36  + 3a. 

36.  a'*  -  a*6  -  a6*  +  6*  by  a^  -  2  a6  -f  b\ 

37.  8m*-14m2-18m  +  21  by  4m3  +  6m-7. 

38.  16a*-96a3  +  216a2-216a-h81  by(2a-3)2. 

39.  7a^-6a^-28  +  81a^  +  3a'-25a;*  by  4-3ar^-5a;. 

40.  2a;«-6a;^-a;*-9a:2_^3a._9  ]^y  2a;^-a;  +  3. 

41.  70a-50-a^-37a2  by  6a-5-a3-2a2. 

42.  ar'-81a;^  +  243/-3a;*2/  by  9xy^-\-x^-{-27f-{-3x^y. 

43.  14a;*-23a;  +  6a:«4-6a^-llar'  +  5-12«3 

by  5a;-3ar^  +  2a^-l. 


44  ALGEBRA. 

44.-  4a«-49a^  +  76a2-lG  by  2a-^  +  5a- -  6a  -  4. 

45.  m^  —  6  mV  +  9 m^n^  —  4  w^  by  (m  +  n)(m  —  7i)(m  -\-2n). 

46.  8a2  -  10a6  +  18  ac  -  36^  +  86c  -  Sc^  by  2a  -  36  +  5c. 

47.  r^''  —  2/-'  +  2  ^/'^^  —  z^'  by  a^^  —  ?/«  -f-  s;'-. 

The   operation  of   division  may  be  abridged  in  certain 
cases  by  the  use  of  parentheses. 

48.  Divide  a?  +{a  —  o  -\-  c)x^  -\-(—ab  —  bc  +  ca)x  —  abc 

by  X  +  a. 

x^+(a  —  b-^c)x^+{  —  ab  —  bc  +  ca)x—abc  \  x -{-  a 

x^  +ax2  \  x^-\-(-b-\-c)x-bc,  Ans. 


(_6  +  c)a:2 

(-6  +  c)x2+(-a&        4-ca)a; 

—  bcx 

—  bcx—abc 

Divide  the  following : 

49.  ar^+(— «  +  6  —  c)a?^-h(— a6  —  6c  + ca)a;4- a6c 

by  X-  +  (—  a  -\-  b)x  —  ab. 

50.  a^  +  (a  +  6  +  c)a^  -f  (a6  +  6c  +  ca)a.-  +  a6c  by  x  +  c. 

51.  a^+(3a-26  -  c)^^ +(- 6a6  +  2  6c  -  3ac)a;  +  6a6c 

by  a;^  +  (3  a  —  c)a7  —  3  ac. 

52.  a(a  +  6)a;2+(a6  +  62-f-6c)a;-c(6-fc)  by  aa;H-(6  +  c) 

53.  m  (m  —  n) x^  -\-  (—  mn  +  n^  —  np)  x -\- p  {n  —p) 

by  maj  —  (n  —  ^). 

54.  a^  +  (a  —  6  —  c)a^  +  (—  a6  +  6c  —  ca)  a?  +  a6c 

by  a:^  —  (6  +  c)  aj  4-  6c. 

55.  a;^  —  (a  +  6  +  c)  x^  -f  (a6  +  6c  +  ca)  a;  —  a6c  by  x  —  a. 

56.  a'(b-c)d-{-a(-b'-{-c'  +  d')-(b  +  c)d 

by  ad  —  (6  +  c). 

57.  d^  -\-  (m  -{-  n)  a  —  2  m^  -\- 11  mn  —  12n^  by  a  —  m  +  4  n. 


DIVISION.  45 

EXAMPLES   FOR   REVIEW. 

62.   1.  Find  the  numerical  value  when   « =  4,   b  =—  7, 
c  =  —  3,  and  d  =  5,  of 

(a  -h  by 


2      c  —  d 


c-^d 

We  have,  (a  +  6)2  =  (4  -  7)(4  -  7)  =  (-  3)(-  3)  =  9, 

and  c-j^-3-5^-8^_^ 

c  +  fZ      -3  +  5        2 

Then,      (a  +  6)2 -^^li?=  9 -(- 4)=  9  +  4  =  13,  Ans. 
c  +  d 

Find  the  numerical  value  of  each  of  the  following  when 
a  =  5,  6  =  —  4,  c  =  —  2,  and  d  =  S: 

2.    {a-b)(b-\-c)(c-d).  3.   b^  -  c- +  2cd  -  d-. 

4.    (a  +  3  6)  (4  c  -  d)  +  (a  -  c)  (2  6  -h  d). 

5.   rt-^-3a26  +  3a62_63.  8.   3a26  _  5  6«c  +  4cU 

g.    Sad     6ab  n    /        i.\'i      /       jn^ 

7    ct  +  2fe     a-56  -^    26a +  236  + 64  c 

'    4:C  +  d     6c-d  '     lla  +  246-7c* 

-J     2a  —  6     36  —  c     4c  —  d 
6— c        c  —  d       d  —  a 

12.   Add  9(a  -  6)  -  8(6  -  c),  -  3(6  -  c)  -  7(c  -  d), 
and  4  (c  —  d)  —  5  (a  —  6). 

9(a-6)-   8(6 -c) 

_    3(6-c)-7(c-(?) 
-5(a-6)  +4(c-d) 


4(a-  6)-  11(6  -c)-3(c-(0,  ^«s- 

13.  Add  4a2(a  +  a;)-6(6-^),  -  Sa\a  + x) -2(b  -  y), 

'     and  -7a^(a  +  x)-\-S(b-y). 

14.  Add  18 (;r  -  yf  -  11  (a^  +  yf,  -  9{x-yy +  7 (x -{-  yf, 

and  -  4  (a;  —  y)^  —  5  (a;  +  yy. 


46  ALGEBRA. 

15.  Subtract  5  (a  -  5)  -  8  (c  +  d)  from  2  (a  -  6)  -  o(c  +  d). 

16.  Multiply  3  (a;  +  2/)  -  5  by  3  (x  4-  2/)  +  5. 

17.  Multiply  7  (a  -  6)  +  4  by  9  (a  -  6)  -  8. 

18.  Divide  6  (m  +  ^)2  -  (m  -h  w)  -  15  by  3  (?w,  +  ^0  -  5. 

19.  Divide  (x-yf  +  l  by  (a;-?/)  +  l. 

20.  Add  |a  +  |ft-ic  and  ia-|6-hfc. 

21.  Add   4a-f6  +  fc  and  ^aH-|6-fc. 

22.  Add  |a;-|?/-y2^2;  and   -ix-j-^y-^z. 

23.  From  i  a  -  f  />  +  |  c  take  ia-^6-fc. 

24.  Subtract  -y3^x 4- 12/ +  i2;  from   -ix-{-^y-^z. 

25.  Multiply  f  a;^  +  ^  a?  -f  ^9^  by  |  a^  -  f. 

26.  Multiply  ^  a^  -  i  a6  +  tV  ^^  by  ^  a  -  ^  6. 

27.  Divide  -\'-^  +  jh  by  fa^  +  |. 

28.  Divide  |a^ --Ja^ft  +  M«^' -  i^'  by  ^a-^b. 

29.  Multiply  a^^+^ft^  -  a%^+^  by  a^^"^  -  b^-\ 

30.  Divide    a^^m-l  _  ^.3y4n+2    ^y    ic"»    2_^y2n+l_ 

31.  Divide  a'^+''  -  ab^~^  by  aP+^  -  ft^^-i. 

32.  Add  3(a;  +  1)^  -  2(.t  -f  1),  5(x  +  l)-7, 

and  -(x  +  l)2-3(a;4-l)+4. 

33.  From  7(x-{-yy-9x(x-\-y)-h4: 

take  12  (a;  +  2/)^  +  ir(a}  +  ?/)  -  11. 


34.  Simplify  5x-[3x-\x-(7 x-Sx-4:)\ -(9x-5x-2)'\. 

35.  AMj\x'-ix-j\,   -^x'  +  ^^^x-i, 

and  |x2-|x+tV 

36.  Multiply  x^  +  (b  —  c)x  —  bc  by  oj  +  a. 


DIVISION.  47 

37.  Divide  a^m+s  ^3  _  ^252^-5  ^^y  ^m+.i  _  5n-4 

38.  Subtract  j\  a^  -  i  «  +  Jg.  from   f^  a?  +  y\  a  -  3^. 

39.  Multiply  {m-ny -\-2{m -n) +  1 

by  (7?i  —  7i)-  —  2  (m  —  ?i)  +  1. 

40.  Multiply  a*-^"-a"6"  +  52'*  by  a^+^ft^  +  aft^+l 

41.  Simplify  (a  +  5)^  -  2  (a  +  5)  (a  -  6)  +  (a  -  6)2. 


42.  Simplify  a-[2a-(6-6c)- |a-(-2&-5c)-36-cS]. 

43.  Multiply  fa^  -  Ja  -  |  by  f  a^  -  a  -  f. 

44.  Divide  ^a""  -  ^a^ -{■  ^a"  -  ^  hy  ^a?-a-  J. 

45.  Divide  a'  -W-^ah  (a^  -  b^)  4- 10  a^b""  (a  -  b) 

by  (a  +  bf-Aab. 

46.  Divide  12  x^+hj-  ^  _  ^3  ^+4^«-4  _  35  ^+7y5n-6 

by  4  y?^+^if  ^  -I-  5  ic'^+'V"-^ 

47.  Multiply  (a  +  &)  x  -  2  a6  by  a;  +  (a  +  6). 

48.  Divide  (a  -  6)'  -  3  (a  -  bfc  +  3  (a  -  6)  c^  -  c^ 

by  (a-6)-c. 

49.      Divide    X*^  -f  a:-"*//"  _^  y^n    ]^y    ^^m  _^  ^myn  _^  yZn 

50.  Multiply  J  a^  -  f  ax  -  1-  x^  by  |  a^  +  f  aa;  +  J  a;^. 

51.  Multiply  ar^  +  (—  a  +  6)a;  —  ctd  by  x  —  c. 

52.  Multiply  a^^  —  ««  +  x""  by  a;P  —  a;'  +  a;''. 

53.  Divide  |a;^- Jx^  +  fa)-^  by  Ja^H-|a;-|. 

54.  Divide  oi?  -\-  {a  —  b  —  c)x^  -\-  {—  ab  +  be  —  ca)x  -{-  abc 

by  X—  c. 

55.  Simplify  (a;  +  2/  +  ^)  [(x  +  y+  zf  -  3 (xy  -hyz-^-  zx)]. 

56.  Simplify  (a  + 6 +  c)(-a  +  6  + c)  (a-6  4-c)(a  +  6 -c). 


48  ALGEBRA. 


VII.   SIMPLE  EQUATIONS. 

63.  The  First  Member  of  an  equation  is  the  expression  to 
the  left  of  the  sign  of  equality,  and  the  Second  Member  is 
the  expression  to  the  right  of  that  sign. 

Thus,  in  the  equation  2ic  —  3  =  3iC  +  5,  the  first  member 
is  2x  —  8,  and  the  second  member  is  3  aj  -j-  5.     . 

Any  term  of  either  meijiber  of  an  equation  is  called  a 
term  of  the  equation. 

The  sides  of  an  equation  are  its  two  members. 

64.  An  Identical  Equation,  or  Ide^itity,  is  one  whose 
members  are  equal,  whatever  values  are  given  to  the  letters 
involved ;  as  (a  +  b)  (a  —  b)=o?  —  b^. 

65.  An  equation  is  said  to  be  satisfied  by  a  set  of  values 
of  certain  letters  involved  in  it  when,  on  substituting  the 
value  of  each  letter  wherever  it  occurs,  the  equation  becomes 
identical. 

Thus,  the  equation  x  —  y  =  b  is  satisfied  by  the  set  of 
values  a;  =  8,  2/  =  3 ;  for  on  substituting  8  for  x,  and  3  for  y, 
the  equation,  becomes 

8-3  =  5,  or  5  =  5; 
which  is  identical. 

66.  An  Equation  of  Condition  is  an  equation  involving 
one  (^r  more  letters,  called  unknown  qumitities,  which  is  not 
satisfied  by  every  set  of  values  of  these  letters. 

Thus,  the  equation  a;  +  2  =  5  is  not  satisfied  by  every 
value  of  Xy  but  only  by  the  value  aj  =  3: 

An  equation  of  condition  is  usually  called  an  equation. 


SIMPLE   EQUATIONS.  49 

67.  If  an  equation  contains  but  one  unknown  quantity, 
any  value  of  the  unknown  quantity  which  satisfies  the 
equation  is  called  a  Root  of  the  equation. 

Thus,  3  is  a  root  of  the  equation  x  +  2  =  5. 
To  solve  an  equation  is  to  find  its  roots. 

68.  A  Numerical  Equation  is  one  in  which  all  the  known 
numbers  are  represented  by  Arabic  numerals;  as, 

2x-17=zx-5. 

69.  A  monomial  is  said  to  be  rational  and  integral  when 
it  is  either  a  number  expressed  in  Arabic  numerals,  or  a 
single  letter  with  unity  for  its  exponent,  or  the  product  of 
two  or  more  such  numbers  or  letters. 

Thus,  3,  a,  and  2  a%c^  are  rational  and  integral. 

70.  If  each  term  of  an  equation,  involving  but  one  un- 
known quantity  x,  is  rational  and  integral,  and  no  term  con- 
tains a  higher  power  of  x  than  the  first,  the  equation  is  said 
to  be  of  the  first  degree. 

us,     X  —     —     I        equations  of  the  first  degree, 
and         d'x  +  h^=c)  ^ 

A  Simple  Equation  is  an  equation  of  the  first  degree. 

PROPERTIES   OF  EQUATIONS. 

71.  It  follows  from  §  9,  2  and  3,  that : 

1.  The  same  number  may  be  added  to,  or  subtracted  from, 
both  members  of  an  equation,  without  destroying  the  equality. 

2.  Both  members  of  an  eqhation  may  be  multiplied,  or 
divided,  by  the  same  number,  without  destroying  the  equvvdty. 

72.  Transposition  of  Terms. 

A  term  may  be  transposed  from  one  member  of  an  equation 
to  the  other  by  changing  its  sign. 


50  ALGEBRA. 

Let  the  equation  be    x  -{-  a  =  b. 
Subtracting  a  from  both  members  (§  71,  1),  we  have 
x=  b  —  a. 

In  this  case,  the  term  +  a  has  been  transposed  from  the 
first  member  to  the  second  by  changing  its  sign. 
Again,  consider  the  equation 

X  —  a  =  b. 
Adding  a  to  both  members,  we  have 
x  =  b  -\-a. 

In  this  case,  the  term  —  a  has  been  transposed  from  the 
first  member  to  the  second  by  changing  its  sign. 

73.  It  follows  from  §  72  that 

If  the  same  term  occurs  in  both  members  of  an  equation 
affected  with  the  same  sign,  it  may  be  cancelled. 

74.  The  sign  of  each  term  of  an  equation  may  be  changed 
without  destroying  the  equality. 

Let  the  equation  be   a  —  a?  =  6  —  c.  (1) 

Transposing  each  term  (§  72),  we  have 

—  b-\-c=  —  a-\-x. 

That  is,  x  —  a  =  c  —  b] 

which  is  the  same  as  (1)  with  the  sign  of  each  term  changed. 

SOLUTION  OF  SIMPLE  EQUATIONS. 

75.  1.  Solve  the  equation ' 

Bx  —  1  =  ^x-\-l. 

Transposing  3  a;  to  the  first  member,  and  —  7  to  the  second,  we 
have 

5x-3a;  =  7  +  l. 

Uniting  similar  terms,  2  re  =  8. 


SIMPLE  p:quations.  51 

Dividing  both  members  by  2  (§  71,  2),  we  have 

X  =  4,  Ans. 

From  the  above  example,  we  derive  the  following  rule : 
Transpose  the  unknown  terms  to  the  first  member,  and  the 

known  terms  to  the  second. 

Unite  the  similar  terms,  and  divide  both  members  by  the 

coefficient  of  the  unknown  quantity. 

2.    Solve  the  equation 

14-5a;=19  +  3a;. 
Transposing,  —  5  a;  --  3  x  =  19  -  14. 

Uniting  terms,  —  8  x  =  5. 

Dividing  by  -  8,  x  =  -  -,  Ans. 

8 

Note  1.     The  result  may  be  verified  by  putting  x  =  —  -  in  the 
given  equation ;  thus, 


U-5(-|)  =  ,9  +  3(-|). 


That  is,  14  +  2^=19-1^. 

8  8 

Or,  137  ^  137     ^^^^^  ^  identical. 

8         8 

EXAMPLES. 
Solve  the  following,  in  each  case  verifying  the  answer : 

3.  9a;  =  7a;  +  28.  10.    7a;  -  29  =  16a;  -  17. 

4.  8a;-5=-61.  11.    13  -  6a;=  13a;  -  6. 

5.  6a;  +  ll  =  a;  +  31.  12.    19  -  16a;  =  27 -28a;. 

6.  9a;-7  =  3a;-37.  13.   9a;- 23  =  20a;  -  18. 

7.  4a;-3  =  8a;-|-33.  14.   30  +  17a;=  27a;  + 22. 

8.  12-13a;  =  6-10a;.  15.   24a;  -  11  =  28  +  11a;. 

9.  5a;H-9  =  14-2a;.  16.   33 a;  +  25  =  41  +  51  a;. 

17.   14a;-|- 21-35= -29a;  +  44a;-22. 


52  ALGEBRA. 

18.  32x-S9  =  25x-10x-Ul. 

19.  12a.'-23a5-h55  =  15aj-75. 

20.  Solve  the  equation 

(2x-  ly  =  2(a;  +  3)  (2a;  -  3)  -  3(6 a;  -  1). 

Expanding  (Note  2),    4  ^2  -  4  x  +  1  =  4  a;2  +  6  x  -  18  -  18  ic  +  3. 
Transposing, 

4x2 -4x- 4x2 -6x4-  18x=  -  18  +  3-1. 
Uniting  terms,  8  x  =  -  16. 

Dividing  by  8,  x=  —2,  Ans. 

Note  2.  To  expand  an  algebraic  expression  is  to  perform  the 
operations  indicated. 

Solve  the  following  equations : 

21.  2(5x  +  l)-4.=:3(x-7)-16. 

22.  10aj-(3cc  +  2)=9a;-(5x-4). 

23.  8a;-5(4a;4-3)=-3-4(2a;-7). 

24.  5.T-6(3-4a.-)=x-7(4-f  .t). 

25.  6x(3x-5)-\-Ul  =  2x(9x-{-l)-{-lS. 

26.  19-5a?(4a;-M)=40-10ic(2a;-r). 

27.  2(4a;-h7)-8(3x-4)  =  6(2a;  +  3)-7(2a;-3). 

28.  (5x  +  7)(3x-8)  =  (5x-\-A)(Sx-5). 

29.  (4.x-7y=(2x-5)(Sx-\-3). 

30.  (5-3a;)(34-4a;) -(7  +  3rc)(l-4ic)= -1. 

31.  (l-3x)2-(a^  +  5)2=:4(i»  +  l)(2a;-3). 

32.  6(4:-xy~5(2x  +  7)(x-2)  =  5-(2x  +  3y. 

PROBLEMS. 

76.  For  the  solution  of  problems  by  algebraic  methods,  no 
general  rule  can  be  given,  as  much  must  depend  upon  the 
skill  and  ingenuity  of  the  student. 


SIMPLE   EQUATIONS.  53 

The  followiug  suggestions  will,  however,  be  found  of 
service : 

1.  Represent  the  unknown  quantity,  or  one  of  the  un- 
known quantities  if  there  are  several,  by  x. 

2.  Every  problem  contains,  explicitly  or  implicitly,  pis- 
cisely  as  many  distinct  statements  as  there  are  unknown  quan- 
tities involved. 

All  but  one  of  these  should  be  used  to  express  the  other 
unknown  quantities  in  terms  of  x. 

3.  The  remaining  statement  should  then  be  used  to  form 
an  equation. 

The  beginner  will  find  it  useful  to  write  out  the  various 
statements  of  the  problem,  as  shown  in  Exs.  1  and  2,  §  77 ; 
after  a  little  practice  he  will  be  able  to  dispense  with  these 
aids  to  the  solution. 

77.  1.  Divide  45  into  two  parts  such  that  the  less  part 
shall  be  one-fourth  of  the  greater. 

Here  there  are  two  unknown  quantities,  the  greater  part  and  the 
less. 

In  accordance  with  the  first  suggestion  of  §  76,  we  will  represent  the 
less  part  by  x. 

The  two  statements  of  the  problem  are,  implicitly  : 

1.  The  sum  of  the  greater  part  and  the  less  part  is  46. 

2.  The  gi-eater  part  is  4  times  the  less  part. 

In  accordance  with  the  second  suggestion  of  §  76,  we  will  use  the 
second  statement  to  express  the  greater  part  in  terms  of  x. 

Thus,  the  greater  part  will  be  represented  by  4  x. 

We  now  in  accordance  with  the  third  suggestion  of  §  76  use  the  first 
statement  to  form  an  equation. 


Thus, 

ix  +  x  =  45. 

Uniting  terms, 

5x  =  45. 

Dividing  by  5, 

X  =  9,  the  less  part. 

Whence, 

4x  =  36,  the  greater  part. 

1 

r     >-     or  THE            A 

UNIVERSITY  J\ 

54  ALGEBRA. 

2.  A  had  twice  as  much  money  as  B ;  but  after  giving  B 
f  35,  he  had  only  one-third  as  much  as  B.  How  much  had 
each  at  first  ? 

Here  there  are  two  unknown  quantities :  the  number  of  dollars 
A  had  at  first,  and  the  number  B  had  at  first. 

Let  X  represent  the  number  of  dollars  B  had  at  first. 

The  first  statement  of  the  problem  is  : 

A  had  twice  as  much  money  as  B  at  first. 

Then  2  x  will  represent  the  number  of  dollars  A  had  at  first. 

The  second  statement  of  the  problem  is,  implicitly : 

After  A  gives  B  ^  35,  B  has  3  times  as  much  money  as  A. 

Now  after  giving  B  $35,  A  has  2  a;  — 35  dollars,  and  B  x -}- 35 
dollars ;   we  then  have  the  equation 

x  +  35  =  3(2x-35). 

Expanding,  aj  +  35  =  6  x  —  105. 

Transposing,  —  5  x  =  —  140. 

Dividing  by  —  5,  x  =  28,  the  number  of  dollars  B  had  at  first ; 

and  2  X  =  56,  the  number  of  dollars  A  had  at  first. 

Note  1.  It  must  be  carefully  borne  in  mind  that  x  can  only  rep- 
resent an  abstract  number;  thus,  in  Ex.  2,  we  do  not  say,  "let  x 
represent  what  B  had  at  first,"  nor  "let  x  represent  the  sum  that  B 
had  at  first,"  but  "let  x  represent  the  number  of  dollars  that  B  had 
at  first." 

3.  A  is  3  times  as  old  as  B,  and  8  years  ago  he  was  7 
times  as  old  as  B.     Required  their  ages  at  present. 

Let  X  =  the  number  of  years  in  B's  age. 

Then,  3  x  =  the  number  of  years  in  A's  age. 

Also,  X  -r-  8  =  the  number  of  years  in  B's  age  8  years  ago, 

and  3  X  —  8  =  the  number  of  years  in  A's  age  8  years  ago. 

But  A's  age  8  years  ago  was  7  times  .B's  age  8  years  ago. 
Whence,  3x-8  =  7(x-8). 

Expanding,  3x  —  8  =  7x  —  56. 

Transposing,  —  4  x  =  —  48. 

Dividing  by  —  4,  x  =  12,  the  number  of  years  in  B's  age. 

Whence,  3  x  =  36,  the  number  of  years  in  A's  age. 


SIMPLE   EQUATIONS.  55 

Note  2.  In  Ex,  3,  we  do  not  say,  "let  x  represent  B's  ag'e,"  but 
"  let  x  represent  the  nxmiber  of  years  in  B's  age." 

4.  A  sum  of  money  amounting  to  $  4.32  consists  of  108 
coins,  all  dimes  and  cents;  how  many  are  there  of  each 
kind  ? 

Let  X  =  the  number  of  dimes. 

Then,  108  —  x=  the  number  of  cents. 

Also,  the  X  dimes  are  worth  10  x  cents. 

But  the  entire  sum  amounts  to  432  cents.  ^  '^.  "V 

Whence,  10  a;  +  108  -  x  =  432. 

Transposing,  9x  =  324. 

Whence,  x  =  36,  the  number  of  dimes  ; 

and  108  —  a;  =  72,  the  number  of  cents. 

PROBLEMS. 

5.  Divide  19  into  two  parts  such  that  7  times  the  less     • 
shall  exceed  6  times  the  greater  by  3. 

'^     6.   What  two  numbers  are  those  whose  sum  is  246,  and 
whose  difference  is  72  ? 

7.  Divide  38  into  two  parts  such  that  twice  the  greater     . 
shall  be  less  by  22  than  5  times  the  less. 

8.  Divide  $22  between  A,  B,  and  C,  so  that  A  may 
receive  $2.25  more  tlian  B,  and  $1.75  less  than  C. 

9.  A  is  5  times  as  old  as  B,  and  in  13  years  he  will  be     . 
only  3  times  as  old  as  B.     What  are  their  ages  ? 

10.  B  is  twice  as  old  as  A,  and  35  years  ago  he  was 
7  times  as  old  as  A.     What  are  their  ages? 

11.  A  had  one-third  as  much  money  as  B;,  but  after  B 
had  given  him  $  24,  he  had  three  times  as  much  money 
as  B.     How  much  had  each  at  first  ? 

12.  A  sum  of  money,  amounting  to  $2.20,  consists  en- 
tirely of  five-cent  pieces  and  twenty -five-cent  pieces,  there 
being  in  all  16  coins.     How  many  are  there  of  each  kind? 


56  ALGEBRA. 

13.  A  is  68  years  of  age,  and  B  is  11.  In  how  many 
years  will  A  be  4  times  as  old  as  B? 

14.  A  is  25  years  of  age,  and  B  is  65.  How  many  years 
is  it  since  B  was  6  times  as  old  as  A  ? 

15.  A  man  has  two  kinds  of  money ;  dimes  and  fifty-cent 
pieces.  If  he  is  offered  $4.10  for  17  coins,  how  many  of 
each  kind  must  he  give? 

16.  Divide  76  into  two  parts  such  that  if  the  greater  be 
taken  from  61,  and  the  less  from  43,  the  remainders  shall 
be  equal. 

17.  What  two  numbers  are  those  whose  sum  is  13,  and 
the  difference  of  whose  squares  is  65  ? 

18.  Find  two  numbers  whose  difference  is  6,  and  the 
difference  of  whose  squares  is  120. 

19.  A  is  14  years  younger  than  B;  and  he  is  as  much 
below  60  as  B  is  above  40.     Kequired  their  ages. 

20.  A  drover  sold  a  certain  number  of  oxen  at  $  60  each, 
and  3  times  as  many  cows  at  $35,  realizing  $  1485  from 
the  sale.     How  many  of  each  did  he  sell  ? 

21.  A  man  has  $4.35  in  dollars,  dimes,  and  cents.  He 
has  one-fourth  as  many  dollars  as  dimes,  and  5  times  as 
many  cents  as  dollars.     How  many  has  he  of  each  kind  ? 

22.  A  garrison  of  4375  men  contains  4  times  as  many 
cavalry  as  artillery,  and  7-^  times  as  many  infantry  as 
cavalry.     How  many  are  there  of  each  kind  ? 

23.  At  an  election  where  5760  votes  were  cast  for  three 
candidates.  A,  B,  and  C,  B  received  5  times  as  many  votes 
as  A,  and  C  received  twice  as  many  votes  as  A  and  B 
together.     How  many  votes  did  each  receive  ? 

24.  Divide  $  115  between  A,  B,  C,  and  D,  so  that  A  and 
B  together  may  have  $  43,  A  and  C  $  65,  and  A  and  D  $57. 


SIMPLE   EQUATIONS.  57 

25.  A  man  divided  $  1656  between  his  wife,  three  daugh- 
ters, and  two  sons.  The  wife  received  4  times  as  much 
as  either  of  the  daughters,  and  each  son  one-third  as 
much  as  each  daughter.     How  much  did  each  receive  ? 

26.  Divide  $  125  between  A,  B,  C,  and  D,  so  that  A  and 
B  together  may  have  $  6o,  B  and  C  $  52,  and  B  and  D  f  54. 

27.  A  man  has  4  shillings  in  three-penny  pieces  and 
farthings;  and  he  has  23  more  farthings  than  three-penny 
pieces.     How  many  has  he  of  each  kind  ? 

28.  Divide  71  into  two  parts  such  that  one  shall  be  4 
times  as  much  below  55  as  the  other  exceeds  37. 

29.  A  square  court  has  the  same  area  as  a  rectangular 
court,  whose  length  is  9  yards  greater,  and  width  6  yards 
less,  than  the  side  of  the  square.  Find  the  area  of  the 
court. 

30.  Two  men,  84  miles  apart,  travel  towards  each  other 
at  the  rates  of  3  and  4  miles  an  hour,  respectively.  After 
how  many  hours  will  they  meet  ? 

31.  Find  three  consecutive  numbers  whose  sum  is  108. 

32.  In  7  years,  A  will  be  3  times  as  old  as  B,  and  8 
years  ago  he  was  6  times  as  old.     What  are  their  ages  ? 

(Let  X  represent  the  number  of  years  in  B's  age  8  years  ago.) 

33.  A  sum  of  money,  amounting  to  $  24.90,  consists  en- 
tirely of  $  2  biUs,  fifty -cent  pieces,  and  dimes ;  there  are  5 
more  fifty -cent  pieces  than  $  2  bills,  and  3  times  as  many 
dimes  as  $  2  bills.     How  many  are  there  of  each  kind? 

34.  Find  two  consecutive  numbers  such  that  the  difference 
of  their  squares,  plus  5  times  the  greater  number,  exceeds 
4  times  the  less  number  by  27. 

35.  Find  four  consecutive  numbers  such  that  the  product 
of  the  first  and  third  shall  be  less  than  the  product  of  the 
second  and  fourth  by  9. 


58  ALGEBRA. 

36.  A  laborer  agreed  to  serve  for  32  days  on  condition 
that  for  every  day  he  worked  he  shoukl  receive  f  1.75,  and 
for  every  day  he  was  absent  he  should  forfeit  ^1.  At  the 
end  of  the  time  he  received  $  28.50.  How  many  days  did 
he  work,  and  how  many  days  was  he  absent  ? 

37.  A  merchant  has  grain  worth  5  shillings  a  bushel, 
and  other  grain  worth  9  shillings  a  bushel.  In  what  pro- 
portion must  he  mix  24  bushels,  so  that  the  mixti^re  may 
be  worth  8  shillings  a  bushel  ?  *)  i 


]4 


38.  A  general,  arranging  his  men  in  a  square,  finds  that 
he  has  43  men  left  over.  But  on  attempting  to  add  1  man 
to  each  side  of  the  square,  he  finds  that  he  requires  108 
men  to  fill  up  the  square.  Kequired  the  number  of  men  on 
a  side  at  first,  and  the  whole  number  of  men. 

39.  In  a  school  of  535  pupils,  there  are  40  more  pupils 
in  the  second  class  than  in  the  first,  and  one-half  as  many 
in  the  first  as  in  the  third.  The  number  in  the  fourth  class 
is  less  by  30  than  3  times  the  number  in  the  first  class. 
How  many  are  there  in  each  class  ? 

40.  A  man  gave  to  a  crowd  of  beggars  15  cents  each,  and 
found  that  he  had  80  cents  left.  If  he  had  attempted  to 
give  them  20  cents  each,  he  would  have  had  too  little  money 
by  10  cents.     How  many  beggars  were  there  ? 

41.  A  tank  containing  120  gallons  can  be  filled  by  two 
pipes,  A  and  B,  in  12  and  15 -iainutes,  respectively.  The 
pipe  A  was  opened  for  a  certain  number  of  minutes ;  it  was 
then  closed,  and  the  pipe  B  opened ;  and  in  this  way  the 
tank  was  filled  in  13  minutes.  How  many  minute's  was 
each  pipe  open  ? 

42.  A  grocer  has  tea  worth  70  cents  a  pound,  and  other 
tea  worth  40  cents  a  pound.  In  what  proportion  must  he 
mix  50  pounds,  so  that  the  mixture  may  be  worth  49  cents 
a  pound  ? 


IMPORTANT   RULES.  59 


VIII.    IMPORTANT  RULES  IN  MULTIPLICA- 
TION AND  DIVISION. 


78.   Let  it  be  required  to  square  a  -\-  b. 


a  +  b 
a  +  b 

a^  -\-ab 

ab  +  b'' 

Whence,  (a  -\- bf  =  a^ -\- 2  ab -\-  b\ 

That  is,  the  square  of  the  sum  of  two  quantities  is  equal  to, 
the  square  of  the  first,  plus  tivice  the  product  of  the  two,  plus 
the  square  of  the  second. 

Example.     Square  3  a  +  ^  be. 

We  have,  (3  a  +  2  &c)2  =  (3  a)2  +  2  x  3  a  x  2  6c  +  (2  hey 
=  9  a2  +  12  a6c  4-  4  h'^c'^,  Ans. 

79.  Let  it  be  required  to  square  a  —  b. 


a- 

-b 

a 

-b 

a' 

-ab 

-  ab  +  b' 

Whence,  (a  -b)-  =  a^-2ab-\-  b\ 

That  is,  the  square  of  the  difference  of  two  quantities  is  equal 
to  the  square  of  the  first,  minus  twice  the  product  of  the  two, 
plus  the  square  of  the  second. 

Example.     Square  4  a;  —  5. 

We  have,  (4  x  -  5)2  =  (4x)2  -  2  x  4ic  x  5  +  52 

=  16  x2  -  40  X  +  25,  Ans. 


60  ALGEBRA. 

80.   Let  it  be  required  to  multiply  a  -{-  b  by  a  —  0. 

a  -\-b 
a  —  b 


o?  +  ab 
—  ab 


Whence,  (a  +  b){a-b)  =  o?  -  b\ 

That  is,  the  product  of  the  sum  and  difference  of  two  quanti- 
ties is  equal  to  the  difference  of  their  squares. 

Example.     Multiply  6  a^  +  6  by  6  a^  —  6. 

We  have,  (6  a^  +  h)  (6  a^-l))  =  (6  a'^y  -  h'^  =  36  a*  _  &2^  j^^s. 

81.  In  connection  with  the  examples  of  the  present 
chapter,  a  rule  for  raising  a  monomial  to  any  power  whose 
exponent  is  a  positive  integer  will  be  found  convenient. 

Let  it  be  required  to  raise  5  a^b^c  to  the  third  power. 

We  have,  (5  a'b'cf  =  5  a'b'c  x  5  a'b'c  x  5  a'b'c  =  125  a^6V. 

We  then  have  the  following  rule : 

Maise  the  numerical  coefficient  to  the  required  power,  and 
multiply  the  exponent  of  each  letter  by  the  exponent  of  the 
required  power. 

EXAMPLES. 

82.  Find  by  inspection  the  values  of  the  following : 

1.  (aj  +  4)2.  9.  (8  +  3m37i2)2. 

2.  (a -3)1  10.  (ab'  +  2a'by. 

3.  (6a- 56)2.  II  {(dxy-lxzf. 

4.  {2xy+^)\  12.  (4>  +  116c)l               ; 

5.  (3m  +  4ti)(3m-4?i).     13.  {9fxf  +  2z'){^xf-2z^). 

6.  (1-2  ay.  14.  {lab-^cd)\ 

7.  (5x2  +  8)(5ar^-8).  15.  (6a^  +  ll/)(6aj5- 11/). 

8.  {a'-Qaf,  16.  (Qa^  +  Sa^. 


IMPORTANT   RULES.  61 

17.  (7m^+12  7i)(77n^-12n).        20.    (3  a*"  +  4  6'*)2. 

18.  (Sx'  +  Ty^y.  21.    {5xP-Sj:^y. 

19.  (13  a'x- 6  byy.  22.    (a^ -^  a')  {a^  -  a"-). 

23.  Multiply  a  +  b-\-chya  +  b  — c. 

ia+  b  +  c)(a-^  6  -c)  =  [(a  +  6)+c][(a  +  6)-c]- 

=  (a  +  6)2-c2  (§80) 

=  a2  +  2  aft  +  62  _  c2,  ^us.        (§  78) 

24.  Multiply  a  +  6  —  c  by  a  —  6  -h  c. 

(a  +  &  -  c)(a  -  ft  4-  c)  =  [a  +(6  -c)][a  -(ft  -  c)] 

=  rt-2_(5_c)2  (§80) 

=  rt2_(62_26c  +  c2)  (§79) 

=  a-  -  b^  +  2bc-  c2,  ^ns. 
Expand  the  following : 

25.  (a  +  ft  +  c)(a-ft4-c).    28.  (a'' -\-a-l){a^  -  a-{-l). 

26.  (aj-2/-f  2)(a?-?/-2).    29.  (x' -\- x  -  2)  (x"  -  x  -  2). 

27.  (a  4-  6  +  c)(a  -  6  -  c).     30.  (1  +  a  +  6)  (1  -  a  -  6). 

31.  (x'-{-2x-hl)(x''-2x-\-l). 

32.  (a  +  2ft-3c)(a-2ft-h3c). 

33.  (a^-f  aft4-?^')(a2-«ft  +  ft2). 

34.  (3x-{-Ay-\-2z){3x-4:y-2z). 


\.  We  find  by  multiplication  : 

x  +  5 

x-5 

x  +  3 

x-3 

a^-\-5x 

x'-Bx 

+  3a;  +  15 

-3x4-15 

x2  +  8a;  +  15     • 

x'-Sx-^-W 

x-f-5 

x-5 

x-3 

x-\-3 

x^-\-5x 

a^-5x 

-3x-W 

+  3a;-15 

a;2  +  2a;-15  x'-2x-15 


62  ALGEBRA. 

In  these  results  it  will  be  observed  that : 

I.  The  coefficient  of  x  is  the  algebraic  sum  of  the  second 
terms  of  the  multiplicand  and  multiplier. 

II.  The  last  term  is  the  product  of  the  second  terms  of 
the  multiplicand  and  multiplier. 

By  aid  of  the  above  laws,  the  product  of  any  two  binomial r 
of  the  form  x  -\- a,  x  -\-h  may  be  written  by  inspection. 

1.  Required  the  value  of  (x  +  8)  (x  —  5). 
The  coefficient  of  x  is  +  8  —  5,  or  3. 

The  last  term  is  8  x  (-  5),  or  -  40. 

Whence,  (a;  +  8)  (a;  -  5)  =  ic^  +  3  x  -  40,  Ans. 

2.  Required  the  value  of  (a  —  6  —  3)  (a  —  6  —  4). 
The  coefficient  of  a  —  &  is  —  3  —  4,  or  —  7. 

The  last  term  is  (-  3)  x  (-  4),  or  12. 

Whence,  {a-h-  3)(a  -  6  _  4)  =  («  -  6)2  -  7  (a  -  5)  +  12,    Ans. 

EXAMPLES. 

Find  by  inspection  the  values  of  the  following : 

3.  (a;  +  6)  (ic  +  4).  14.  (a  +  6  -  7>(a  +  6  +  8). 

4.  {x-2)(x  +  Z).  15.  (x-^a){x-na). 

5.  (a;  -  10)  (a;  -  1).  16.  {x -ir  y)  (x  -  2  y). 

6.  (x-}-5)(a^-6).  17.  (a  +  lU)(a-66). 

7.  (a  +  l)(a  +  9).  18.  {a  +  1  x){a  +  ^x). 

8.  (a-7)(a  +  4).  19.  {x -y  -  4.){x- y +  10). 

9.  (m  +  5)(m-l).  20.  {x -llz){x +  9z). 

10.  {x'-7){x'-2).  21.  {x^  +  3y){^  +  ^y). 

11.  (ti^  +  3)  (71^  _  10).  22.  (aT^-9m2)(a^-6m2). 

12.  (a6  4- 2)  (a6  +  11).  23.  {ah +  Scd)  {ah -12  cd). 

13.  {xy-12){xy-S).  24.  (x  + y  +  12)(aj  +  ^  -  9). 


IMPORTANT   RULES.  63 

84.  We  have  by  §  80, 

=a  —  h,  and  =  a-\-h. 

a  -\-b  a  —  b 

That  is,  if  the  difference  of  the  squares  of  two  quantities  he 
divided  by  the  sum  of  the  quantities,  the  quotient  is  the  dif- 
ference of  the  quantities. 

If  the  difference  of  the  squares  of  tivo  quantities  be  divided 
by  the  difference  of  the  quantities,  the  quotient  is  the  sum  of 
the  quantities. 

1.   Dividel6a'6^-9by  4a62-h3. 

We  have,  16  a^b*  =  (4  ab'^y.  (§  81 ) 

Whence,  \6a'^b*-9  ^  4  ^52  _  3,  Ans. 

4  ab-^  +  3 

EXAMPLES. 
Write  by  inspection  the  values  of  the  following : 

2  ^-^  5    25  <^'- ^6  8     1-64  nv'n^ 

'    x-\-l'  '     5a^-{-6  '  '      1+8 mw" 

3  ^-^'  6    ^^'-^^y'  9    4ci^6^-c^ 

'    2-a  '      3x-\-iy  '  '    2ab'-c'' 

•4     16  m- -49  y     2Da--b\  ^^    49  m-- 100  u" 


4m-7  5a-b'  7m-10n^ 

..     81/-196a;^  ,3     144x-y  -  169^*^ 

■  9?/-hl4a^  *  ■      12xy'-lSz'  ' 

12     121&V-64a'^  ^^    225a^Q-64  6^¥ 

■  llbc  +  Sa     '  '       15a^-h8  6V 

85.  We  find  by  actual  division  : 

t±^  =  a^-ab  +  b^ 
a  +  b 

and  9LJ=L^=a'^ab-\-b\ 

a  —  b 


64  ALGEBRA. 

That  is,  if  the  sum  of  the  cubes  of  two  quantities  he  divided 
by  the  sum  of  the  quantities,  the  quotient  is  the  square  of  the 
first  quantity,  minus  the  product  of  the  two,  plus  the  square  of 
the  second. 

If  the  difference  of  the  cubes  of  two  quantities  be  divided  by 
the  dijfference  of  the  quantities,  the  quotient  is  the  square  of  the 
first  quantity,  plus  the  product  of  the  two,  plus  the  square  of 
the  second. 

1.  Divide  1  -f-  8  a^  by  l-f  2  a. 

We  have,  l±^^l^{2aY 

=  1  -2a  +  (2a)2 
=  1  -  2  a  +  4  a2,  Ans. 

2.  Divide  27  3^-64 /by  3  iK- 4?/. 

We  have,  27^^-64^  ^  {^xY-j^vY 

3x  —  4y  Sx  —  4:1/ 

=  (3a;)2  +  (3x)(4?/)  +  (4i/)2 
=  9x^  +  l2xy  -\-  16^/2,  Ans. 


EXAMPLES. 
Find  the  values  of  the  following : 

3    «''-!                    8    ?L^^Z^.  13    8a;^-125/  ' 

a^b  —  (?  2x  —  hy'^ 

g     1  +  64  m\  ^^    a%''  +  512  & 

l-|-4m  '       a&  +  8c' 

5    !!^!_±§.                10    ?lizi^.  15    64  mhv'  +  343 

6  —  X  4  mn  +  7 

^^     g^  +  125  ^g    729a^-125a-''^ 

a  4-  5  9  a^  —  5  £c 

^     a;«  +  /'                 .g    l-343a^6^  ,«    512ajy  +  27/ 

x^^f                    '      l-7ab'  '  '       Safy-\-3z'    ' 


a'-l 

a-1 

1  +  a^ 

l-\-x 

m^  +  8 

m-f  2 

27  -  a'' 

3-a 

a^«  +  /' 

IMPORTANT   RULES.  65 


86.   We  find  by  actual  division : 


a-\-b 


a'b  4_  ab-  -  h\ 


^!i^I±=:a^  +  a'b  +  ab'  +  b^ 
a  —  b 

a^-\-b^ 


a-\-b 
a'-b' 


=  a'-  cv'b  +  d'b-  -  aW  +  b\ 

=  a*  -h  o?b  -h  d-b-  -\-  aW  +  b^ ;  etc. 


In  these  results  we  observe  the  following  laws : 

I.  The  number  of  terms  is  the  same  as  the  exponent  of 
a  in  the  dividend. 

II.  The  exponent  of  a  in  the  first  term  is  less  by  1  than 
its  exponent  in  the  dividend,  and  decreases  by  1  in  each 
succeeding  term. 

III.  The  exponent  of  b  in  the  second  term  is  1,  and 
increases  by  1  in  each  succeeding  term. 

IV.  If  the  divisor  is  a  —  b,  all  the  terms  of  the  quotient 
are  positive ;  if  the  divisor  is  a-\-b,  the  terms  of  the  quo- 
tient are  alternately  positive  and  negative. 

87.  The  following  principles  are  of  great  importance. 
If  n  is  any  positive  integer,  it  will  be  found  that : 

I.  a"  —  6"  is  always  divisible  by  a  —  b. 

Thus,  a^  —  b^,  a^  —  b^,  a^  —  b\  etc.,  are  divisible  hj  a  —  b. 

II.  a"  —  6"  is  divisible  by  a  -\-  b  if  7i  is  even. 

Thus,  a^  —  b^,  a^  —  b*,  a^  —  b%  etc.,  are  divisible  by  a  +  6. 

III.  a**  -f  6"*  is  divisible  by  a-\-b  ifn  is  odd. 

Thus,  a^  +  b%  a^  +  b^,  a'  +  b'^,  etc.,  are  divisible  by  a  -f  6. 

IV.  a"  +  b"  is  divisible  by  neither  a  -\-bi  nor  a  —  b  if  n 
is  even. 

Thus,  a^  -h  6^,  a*  +  b^,  cf  +  b^,  etc.,  are  divisible  by  neither 
a  -f  6  nor  a  —  b. 


5  ALGEBRA. 

88.   1.   Divide  cd  -  h'  by  a  —  b. 

Applying  the  laws  of  §  86,  we  have, 


^6  _|.  a^l)  ^  a^l)2  _j.  Qj3^3  +  (^2^4  ^  ^55  ^  56^  ^,^5. 


a  —  b 
2    Divide  16 ic^  -  81  by  2x  +  3. 

We  have,     16^-81^(2^)4-34 
2  X  +  3  •        2  a;  +  3 


(2  a:)3  -  (2  :r)2  x  3  +  (2  a;)  x  32  -  3^ 
:8x3  -  12^2+  18iC-27,  ^?is. 


EXAMPLES. 

Find  the  values  of  the  following : 


a' 

-1 

a 

+  1 

X' 

-1 

X 

-1 

a« 

-b' 

a 

-hb 

1- 

-x' 

1- 

-X 

a« 

-b' 

0? 

-b' 

x^': 

f  +  z^' 

g 

16 -x' 

2-x 

10. 

l-16a^ 

l  +  2a 

11. 

a'  +  b'  ■ 
a-\-b 

12. 

1-m' 

1  —  m 

13. 

32 +  a^ 

2  +  a 

14 

m^  —  71^ 

15. 
16. 

18. 


a^y  -\-^ 


m  +  n 


20. 


64a«-6« 

2a-& 

81  a;^- 2/' 

3a;  +  2/  ■ 

a^-243a^ 

a  —  3x 

81  a^- 256  6^ 

3a-46 

243  0^^  +  32?/^ 

3a;  +  22/ 

128m^-n" 

2m  —  Ti^ 


FACTORING.  67 


IX.    FACTORING. 

89.  To  Factor  an  algebraic  expression  is  to  find  two  or 
more  expressions  which,  when  multiplied  together,  will 
produce  the  given  expression. 

90.  A  Common  Factor  of  two  or  more  expressions  is  an 
expression  which  will  exactly  divide  each  of  them. 

91.  A  monomial  can  always  be  factored ;  thus, 

12  a^bc-  =  2x2xSxaxaxaxbxcxc. 

It  is  not  always  possible  to  factor  a  polynomial ;  but 
there  are  certain  types  which  can  always  be  factored,  the 
more  important  of  which  will  be  taken  up  in  the  present 
chapter. 

92.  Case  I.  When  the  terms  of  the  expression  have  a 
common  monomial  factor. 

1.   Factor  Uxy^-353^y\ 

Each  term  contains  the  monomial  factor  7  xi/^. 

Dividing  the  expression  by  7  xy^^  the  quotient  is  2y^  -  6  y?-. 

Whence,       14  x^  -  35  y?t  =  7  xy^(2  y^  -  5  x^) ,  Ans. 

EXAMPLES. 
Factor  the  following : 

2.  a3  +  4a.  7.  12 a^  -  20 a'^  +  4 a^. 

3.  6a5^-14aj3.  8.  a^6 V  +  a^^V  +  a'6c«.        ^ 

4.  30wi2-5m3.  9.  12  a^^^  +  24  a;/ -  42  a;y. 

5.  l^a^h^  +  ^ah'^  ■  10.  14  a«6*  +  21  a^d^  _  49  ^352 

6.  56a^2/'-32a;y.  11.  81  m^n  +  54  mV  +  9  m V. 

12.  48  a^y- 144x3/  + 108  a^^. 

13.  70  aV  -  126  aV  ^  112  aV. 


68  ALGEBRA. 

93.  Case  II.  When  the  expresdon  is  the  sum  of  tivo  bino- 
mials which  have  a  common  binomial  factor. 

1.  Factor  ac  —  bc^  ad  —  bd.  • 

By  §  92,  {ac  -  be)  +  {ad  -  bd)  =  c{a  -  6)  +  d{a  -  b). 
The  two  binomials  have  the  common  factor  a  —  b. 
Dividing  the  expression  by  a  —  6,  the  quotient  is  c  +  d. 
Whence,       ac  —  6c  +  ad  —  bd  =  («  —  &)  (c  +  d),  Aiis. 

If  the  third  term  of  the  given  expression  is  negative,  as 
in  the  following  example,  it  is  convenient  to  enclose  the 
last  two  terms  in  a  parenthesis  preceded  by  a  —  sign. 

2.  Factor6ic3-15x2-8ic  +  20. 

6^3-  15x2 -8x  + 20  =(6x3-  15x2)- (8x  -  20) 
zz3x2(2x-5)-4(2x-5) 
=  (2x-5)(3x2-4),  Ans. 

EXAMPLES. 

Factor  the  following : 

3.  ab  +  an  +  bm  +  mn.  9.  3  x"'  +  6  x^  +  ic  +  2. 

4.  ax  —  ay  -{-bx  —  by.  10.  10  mx  — 15  nx  —  2  m  -f  3  n. 

5.  ac  —  ad  —  bc-\-  bd.  11.  a^x  +  abcx  —  a^by  —  b^cy. 

6.  a3  +  a2  +  (*  +  l.  12.  a'bc  -  ac'd -{- abH  -  bed'.  ^ 

7.  4a^-5a;2-4a;-f5.  13.  30a2- 12^^  -  55a  +  22. 

8.  2  +  3a-8a2-12a^  14.  56  -  32  a^  +  21  a^  -  12  ic^. 

15.  a^b^ -h  a'^bcd' -\- ab'c'd -\- cH^ 

16.  Sax  —  ay  —  9bx-{-S by. 

17.  4  a^  +  x^y'  —16xy  —  4:y^. 

18.  20 ac  +  Wbc  +  Aad-^Sbd. 

19.  16  mx  —  56  my  +  10  7ix  —  35  ny. 

20.  45  a^  -  20  a^ft^  _  53  ^  J  ^  28  6^. 


FACTORING.  69 

94.  If  an  expression  can  be  resolved  into  two  equal 
factors,  it  is  said  to  be  a  perfect  square,  and  one  of  the 
equal  factors  is  called  its  square  root. 

Thus,  since  9  a^6^  is  equal  to  3  a%  x  3  a^b,  it  is  a  perfect 
square,  and  3  a^b  is  its  square  root. 

Note.  9  a4&-^  is  also  equal  to  (  -  3  a^b)  x  (  -  3  a^b) ;  so  that  -  3  a'^b 
is  also  its  square  root.  In  the  examples  of  the  present  chapter,  we 
shall  consider  the  positive  square  root  only. 

95.  The  following  rule  for  extracting  the  square  root  of 
a  perfect  monomial  square  is  evident  from  §  94 : 

Extract  the  square  root  of  the  numerical  coefficient,  and 
divide  the  exponent  of  each  letter  by  2. 
Thus,  the  square  root  of  25  a*6V  is  5  a^b^c. 

96.  It  follows  frouL  §§78  and  79  that  a  trinomial  is 
a  perfect  square  when  its  first  and  last  terms  are  perfect 
squares  and  positive,  and  the  second  term  twice  the  product , 
of  their  square  roots. 

Thus,  in  the  expression  4:X^  —  12xy  -\-9y^,  the  square 
root  of  the  first  term  is  2  x,  and  of  the  last  term  3  y ;  and- 
the  second  term  is  equal  to  2(2x)(Sy). 

Whence,  4:X^  — 12xy-{-9y^  is  a  perfect  square. 

97.  To  find  the  square  root  of  a  perfect  trinomial  square, 
we  simply  reverse  the  rules  of  §§  78  and  79 : 

Extract  the  square  roots  of  the  first  and  last  terms,  and 
connect  the  results  by  the  sign  of  the  second  term. 

Thus,  the  square  root  of4ar'  —  12a^-f9y^is2a;  —  3?/. 

98.  Case  III.  When  the  expression  is  a  perfect  trinomial 
square  (§  96). 

1.   Factor  a^-\-2ab^+b\ 

By  §  97,  the  square  root  of  the  expression  is  a  +  b^. 

Whence,      a^ -\- 2  ab'^  +  6*  =(a  +  b'^y-  =  (a  +  b"^)  (a  +  b"-),  Ans. 


70  ALGEBRA. 

2.    Factor  25  x^  -  40  xy""  +  16  y\ 

The  square  root  of  the  expression  is  5x  —  4?/3. 

Whence,  25 x2  -  40 xy^  -\- \Qy^  ^{px  -  ii/y 

Note.     The  expression  may  be  written  16  y^  —  40  xy^  +  25  ic^  ;   in 
which  case,  according  to  the  rule,  its  square  root  is  4«/3  —  5ic. 
Thus,  another  form  of  the  result  is 

162/6  _  40iC2/^  +  25x2  =(4 «/3  -  5a:)(4y3  _  5^)^ 

EXAMPLES. 

Factor  the  following : 

3.  wr  +  2mn  +  n\  15.  Ua'b^ +  lQahcd  +  (?d\ 

4.  o?-2ah  +  h\  16.  100a^-60a^4-9x^ 

5.  9  +  6flJ  +  a;2.  17.  49 m*  +  112 m^ii^  +  64 ti^. 

6.  a2-8a  +  16.  18.  121  a^fts ^  132 ^^c^  +  36 c^ 

7.  49a.-2  +  14a;2/  +  2/2.  19.  144i»y-120a;y+25icy. 

8.  m''-l()mn  +  2bn\  20.  64 a^fts.^  175 ^52^ .1.12162^2 

9.  4  a*  -  4  a'hh  +  6V.  21.  49a^2/'  -  168iC2/«^  +  144^^  ^' 

10.  mV  +  18ma;4-81.  22.  36aV-156aV+169a2a!4. 

11.  4a2  -  20aa;  +  25a^.  23.  (a  +  6)2  _  4  (a  +  6)  +  4. 

12.  9a2  +  42a6+4962.  24.  (x  - 2/)' + 10 (a;  -  ^)  +  25. 

13.  81  a^- 72  0^2/ +  16  2/'.  25.  16(a  +  a;)2  + 8(a  +  a^)  +  l. 

14.  a^  +  12  x^yz  +  36  yh\  26.  4(a  -  hf  -  12(a  -  6)  +  9. 

99.   Case  IV.     When  the  expression  is  the  difference  of  tivo 
perfect  squares. 

By  §  80,  aP  -h'' =  {a  +  h)(a  ~  h). 

Hence,  to  obtain  the  factors,  we  reverse  the  rule  of  §  80 : 

Extract  the  square  root  of  the  first  square,  and  of  the  second 

square ;  add  the  results  for  one  factor,  and  subtract  the  second 

result  from  the  first  for  the  other. 


FACTORING.  71 

1.    Factor  36  a2  -  49  6\ 

The  square  root  of  36  a^  is  6  a,  and  of  49  M  is  7  6^. 

Whence,     36  d^  -  49  6^  =  (6  a  +  7  62)  (6  a  -  7  62) ,  ^  ns. 

EXAMPLES. 

Factor  the  following : 

2.  a'-W.  8.  49m*-16ii2.  14.  144mV-49. 

3.  x^-l.  9.  25  a^- 64  6V.  16.  36a«-169a:». 

4.  9 -ml  10.  100a^/-92*.  16.  Ux'^-imfz\ 

5.  KSx'-f.  11.  64m*-81n«.  17.  64 a^^^s _  225 c>«. 

6.  4.  a' -25.  12.  121  a^fo^ -  4  c^d^.  18.  169-144a^y^ 

7.  l-36a26l  13.  81x«-100/.  19.  196aV-1216y. 

20.  Factor  (2x-Syy-(x-  y)\ 
We  have,    (2x  —  ^yy-{x-yY 

=  l(2x-Zy)-V{x-y)^[{2x-^y)-{x-y)] 
=  (2x-Sy  +  X  -y){2x-  Sy  -x  +  y) 
=  {Sx-iy)(x-2y),  Ans. 

Factor  the  following : 

21.  (a  -f  by  -  c\  28.  (a  +  hf  -  (c  -  d)\     ^ 

22.  {m-nf-x'.  29.  {a  -  xf  -  (h  -  yf. 

23.  d'-ih-cf.   ^  30.  (x  +  yy-{m  +  nf. 

24.  aj2_(2,  +  2)2.  31.  (8  a -5)2 -(3  a +  7)2. 

25.  m2-(t»-p)2.  32.  (4  a; +  1)2 -(a; +  6)2. 

26.  {lx-2yf-y\  33.  (7a  -  56)2  -  (5a- 26)2. 

27.  {a-bf-{x  +  y)\  34.  (9  a;  +  82/)'- (2  a;  -  3?/)2. 

A  polynomial  may  sometimes  be  expressed  in  the  form 
of  the  difference  of  two  perfect  squares,  when  it  may  be 
factored  by  the  rule  of  Case  TV. 


72  ALGEBRA. 

35.  Factor  2  nm  +  m^  —  1  +  n^ 

Since  2  mn  is  the  middle  term  of  a  perfect  trinomial  square  whose 
first  and  third  terms  are  w^  and  n'  (§  96),  we  arrange  the  given  ex- 
pression so  that  the  first,  second,  and  last  terms  shall  be  grouped 
together,  in  the  order  m^  +  2  mn  +  n^  ;  thus, 

2  wm  -h  w2  -  1  +  w2  =  (w2  +  2  mn  +  n^)  -  1 

=  (m  +  w)2  -  1,  by  Case  III. 
=  (m  +  w -I- l)(m  +  w  —  1),  JlWS. 

36.  Factor  12 1/  +  aj^  -  9  2/^  -  4. 

We  have,  I2y  +  x^  -  9y^  -  ^  =  x^  -9y^  -^  12y  -  4: 

=  x2-(9?/2_  12  2/ +  4) 
=  x'^  -  (3  ?/  -  2)2,  by  Case  III. 
=  lx  +  iSy-2)][x-iSy-2)^ 
=  (x-}-Sy-2){x-3y-\-2),Ans. 

37.  Factor  a2-c2  +  62_(^2_2cd-2a6. 

We  have,     a^  -  c^  -}-  b^  -  d^  -  2cd  -  2  ab 

=  a^-2ab-^b'^-c^-2cd-d^ 
=  (a2  -  2  a6  +  62)  _  (c2  +  2  c(Z  +  c?^) 
=  (a  -  6)2  -  (c  +  d)2,  by  Case  III. 
=  [(a  -  6)  +  (c  +  d)J[(a  -  6)  - (c  +  <^)] 
=  {a  —  b  +  c  +  d)(a  —  b  —  c  —  d),  Ans. 

Factor  the  following : 

38.  a2-2a6  +  62_c2.  43.  2  mn  -  jv" -{- 1  -  m\ 

39.  m'-{-2mn-{-n--p\  44.  Oa^  -  24  a5  +  16  6^  _  4c'-, 

40.  o?-x'-2xy-f.  45.  16  a^- 4/  + 20?/^ -25.:' 

41.  a^_2/2_22^22/;2.  46.  4.n^ +  m^ -x" -4.m7i.^\ 

42.  62-44-2a6  +  a2.  47.  4a2- 66-9-62. 

48.  10052/ -92^  + 2/2 +  25a^. 

49.  a2-2a6  +  62-c2-f 20^-^2.'"^ 

50.  a^  -  62  -f  a^  -  /  +  2  aa;  +  2  by. 


FACTORING.  73 

51.   a?  4-  ?yi^  —  \f  —  ii^  —  2  mx  —  2  ny. 

53.  4a2  +  4a6  +  62-9c2  4-12c-4. 

54.  16  /  -  36  -  8  xy-  z^  +  a^  -  12  2. 

55.  m2-9/i2  +  25a2-62-10a??H-66«. 

100.   Case  V.    When  the  expression  is  a  trinomial  of  the 
form  a^  -\-  ax  -\-  b. 
We  have  by  §  83, 

(a; -f.  5)(a;  +  3)  =  ar^ -f  8a;  +  15, 

(x-5){x  _  3)  =  ;r2  _  8x  -f  15, 

(a;  H-  5)  (a;  -  3)  =  a^  +  2  x  -  15, 

and  (a;  -  5)(a;  +  3)  =  ar^  -  2x  -  15. 

In  certain  cases  it  is  possible  to  reverse  the  process,  and 
resolve  a  trinomial  of  the  form  x-  -\-  ax  -{-b  into  two  binomial 
factors. 

The  first  term  of  each  factor  will  obviously  be  x ;  and  to 
obtain  the  second  terms,  we  simply  reverse  the  rule  of  §  83. 

Find  two  numbers  whose  algebraic  sum  is  the  coefficient  of  x, 
and  whose  product  is  the  last  term. 

1.  Factor  ar^ +  14  a;  H- 45. 

We  find  two  numbers  whose  sum  is  14,  and  product  45. 
By  inspection,  we  determine  that  the  numbers  are  9  and  6. 
Whence,        a;^  +  14  x  +  45  =  (x  +  9)  (x  +  5),  Ans. 

2,  Factor  ar^  -  5  a;  +  4. 

We  find  two  numbers  whose  sum  is  —  5,  and  product  4. 
Since  the  sum  is  negative,  and  the  product  positive,  the  numbers 
must  both  be  negative. 

By  inspection,  we  determine  that  the  numbers  are  -  4  and  —  1. 
Whence,  x^  -  5 x  +  4  =  (x  -  4)  (x  -  1),  Ans. 


74 


ALGEBRA. 


3.  Factor  aj2  +  6  a;  -  16. 

We  find  two  numbers  whose  sum  is  6,  and  product  —  16. 

Since  the  sum  is  positive,  and  product  negative,  the  numbers  must 
be  of  opposite  sign,  and  the  positive  number  must  have  the  greater 
absolute  value. 

By  inspection,  we  determine  that  the  numbers  are  -f  8  and  —  2. 

Whence,         x^  -\-  6 x  -  16  =  (x  -\-  S)  {x  -  2),  Ans. 

4.  Factor  cc^  _  a;  _  42. 

We  find  two  numbers  whose  sum  is  —  1,  and  product  —  42. 
The  numbers  must  be  of  opposite  sign,  and  the  negative  number 
must  have  the  greater  absolute  value. 

By  inspection,  we  determine  that  the  numbers  are  —  7  and  +  6. 
Whence,  x^  -  x  -  42  =  (x  -  7)  (cc  +  6),  Ans. 

Note.     In  case  the  numbers  are  large,  we  may  proceed  as  follows  : 

Required  the  numbers  whose  sum  is  —  26,  and  product  —  192. 

One  number  must  be  + ,  and  the  other  — . 

Taking  in  order,  beginning  with  the  factors  +  1  x  —  192,  all  possible 
pairs  of  factors  of  —  192,  one  of  which  is  +  and  the  other  — ,  we 
have : 

+  1  X  -  192, 

+  2  X  -  96, 

+  3  X  -  64, 

+  4  X  -  48, 

+  6  X  -  32. 

Since  the  sum  of  -f  6  and  —  32  is  —  26,  they  are  the  numbers 
required. 

EXAMPLES. 
Factor  the  following : 

5.  aj2  +  6x  +  8. 

6.  cc2  -  13  aj -h  22. 

7.  x'-\-6x-7, 

8.  x'-4:x-21. 

9.  a^-lla;  +  24/' 
10.  a^-\-Sx-20. 


11.  x^-x-6. 

12.  x'  +  lOx-^-d. 

13.  a'-7a-U. 

14.  a''-^a-2. 

15.  m2  + 11m  4-30. 

16.  n'-Tn  +  e, 


FACTORING.  75 

17.  a^  +  3a?-40.  31.  z'—21z +  110. 

18.  2/'+18?/  +  77.  32.  x^  +  17  x^  -  84. 

19.  a2  -  15  a  +  54.  33.  a*  -^2oa'-\-  150.      • 

20.  m2-2m-48.  34.  m«-5m3-36. 

21.  c2h-15c  +  36.  35.  n«  +  10n*-96. 

22.  a^-12ic  +  32.  36.  a^y^ _  ^g ^  _^ 34 

23.  a^-6a;-55.  37.  a'b' -^  2S  ab -h  160. 

24.  n'-\-2n-63.  38.  icy  -  27  a^i/ +  50. 

25.  m2-18??i  +  72.  39.  a^o;*  +  5  a V  -  126. 

26.  a2-3a-70.  40.  mV  -  11  m?i3  -  152. 

27.  x'  +  4.x-96.  41.  (a +  &)'+ 23(a  + 6)+ 60. 

28.  if2  +  24»  +  95.  42.    (a;-2/)=^+3(ic-2/)-180. 

29.  62-106-24.  43.    (a -6)^-22 (a- 6) +112. 

30.  c2  +  20c  +  84.  44.    {x -\- yf  -  2  (x -\- y)  -  US. 

45.  Factor  .t^  +  6  abx  -  27  a^t^. 

We  find  two  quantities  whose  sum  is  6  aft,  and  product  —  27  a^b^. 
By  inspection,  we  determine  that  the  quantities  are  —  Sab  and  9 ab. 
Whence,    x^  ^Qabx-  27  a^fts  =(x  -  3  a6)  (x  +  9  a6),  ^ws. 

46.  Factor  1  _  3  a  -  88  al 

We  find  two  quantities  whose  sum  is  -3  a,  and  product  -  88  a^. 
By  inspection,  we  determine  that  the  quantities  are  8  a  and  -  11  a. 
Whence,       1  -  3rt  -  88a2  =(1  +  8a)(l  -  11a),  Ans. 

Factor  the  following : 

47.  a2  +  12a6-h35  6l  51.  a'i.5am-66m\ 

48.  x^-llax-\-2Sa'.  52.  m^  +  16  w?i  +  48  ^i^. 

49.  ar^  H-  4  xy  —  5y^.  53.  x^  —  mx  —  12  m\ 

50.  l-2a-3a2.  54.  l-14a  +  33al 


76  ALGEBRA. 

55.  a^-4:ab-60  h\.  61.    1  +  18  a5  +  80  a^h\ 

56.  l-\-x-12x\  62.   x'-\-lxy  -my\ 
67.   x"  -  15  xy  +  50  y\  63.   a"})'  +  16  a?>c  +  28  c\ 

58.  i»2  4.  20  aaj  +  99  al  64.    x" -21  xyz +  lQSyh\ 

59.  m2-16mn4-15n^  65.    l+llxy-2&x-y\ 

60.  a^-aft- 20  51  66.   a«  -  6  a-^ftc^  -  160  6  V. 

101.  If  an  expression  can  be  resolved  into  three  equal 
factors,  it  is  said  to  be  a  perfect  cube,  and  one  of  the  equal 
factors  is  called  its  cube  root. 

Thus,  since  27  a%^  is  equal  to  3  a^6  x  3  orb  x  3  d^b,  it  is  a 
perfect  cube,  and  3  a^b  is  its  cube  root. 

102.  The  following  rule  for  extracting  the  cube  root  of  a 
perfect  monomial  cube  is  evident  from  §  101 : 

Extract  the  cube  root  of  the  numerical  coefficient,  and  divide 
the  exponent  of  each  letter  by  3. 

Thus,  the  cube  root  of  125  a^6V  is  5  a^Wc. 

103.  Case  VI.  When  the  expression  is  the  sum  or  differ- 
ence of  two  perfect  cubes. 

By  §  85,  the  sum  or  difference  of  two  perfect  cubes  is 
divisible  by  the  sum  or  difference,  respectively,  of  their 
cube  roots. 

In  either  case,  the  quotient  may  be  obtained  by  aid  of  the 
rules  of  §  85. 

1.  Factor  a^  +  l. 

The  cube  root  of  a^  is  a,  and  of  1  is  1 ;  hence,  one  factor  is  a  +  1. 
Dividing  a^  +  1  by  a  +  1,  the  quotient  is  a^  —  a  +  1  (§85). 
Whence,  a^  +  1  =  (a  +  l)(a2  -  a  +  1),  Ans. 

2.  Factor  27  a^- 64/. 

The  cube  root  of  27  x^  is  3  x,  and  of  64  y^  is  4  tj  (§  102). 
Hence,  one  factor  is  3  x  —  4  y. 

Dividing  27  x»  -  64  y^  by  3  x  -  4  y,  the  quotient  is  9  x^  +  12  xy  +  16  y^ 
(§  85). 

Whence,    27  x^  -  64  2/3  =  (3  x  -  4  y)  (9  x^  +  12  xy  +  16  y^) ,  Ans. 


3. 

m^  -\-  n^. 

9. 

64a^  +  l. 

4. 

a'  -  h\ 

10. 

l-125al 

5. 

a?-l. 

11. 

27a^-8/. 

6. 

a^  -  f7^. 

12. 

8  a^ly"  -h  125. 

7. 
8. 

a^  +  Qi^. 
1  +  m\ 

13. 
14. 

216  -  m\ 
125-64ary.'' 

FACTORING.  77 

EXAMPLES. 
Faxjtor  the  following : 

15.  m«4-343n3. 

16.  125  6^- 216  c^. 

17.  M^m^-]-^x^. 

18.  27a«-f  343  6«. 

19.  512a^4-27yz«. 

20.  64  a«6«  -  729  c». 

104.  Case  VII.  When  the  expression  is  the  sum  or  differ- 
ence of  two  equal  odd  powers  of  ttvo  quantities. 

By  §  87,  the  sum  or  diiference  of  two  equal  odd  powers 
of  two  quantities  is  divisible  by  the  sum  or  difference, 
respectively,  of  the  quantities. 

In  either  case,  the  quotient  may  be  obtained  by  aid  of  the 
rules  of  §  86.  ,  C  y  ST 

i^a    a     -f     7  ^    (^ 

1.    Factor  a^  +  2>^.  I  y  ^     ^      X     "^ 

By  §  87,  one  factor  is  a  +  6. 

Dividing  a^  +  6^  by  a  +  6,  the  quotient  is 

a*  -  a^h  +  aW  -  ab^  +  6*.  (§  86) 

Hence,    n^  -\- b^  =  (a -\-  b)  (a*  -  a^b  +  a^b^  -  ab^  +  6*),  Ans. 


EXAMPLES. 
Factor  the  following : 

2.  x^-f.  6.   aJ-hbl  10.  l4-32ic5. 

3.  a-'  +  l.  7.    1-xl  11.  243m^-l. 

4.  l-m\  8.    m^  +  1.  12.  a:^  -  128. 

5.  .ry4-2'.  9.   32 -a\  13.  32  a* -h  243  6^ 


105.   By  application  of  the  rules  already  given,  an  ex- 
pression may  often  be  resolved  into  more  than  two  factors. 


78  ALGEBRA. 

If  the  terms  of  the  expression  have  a  common  monomial 
factor,  the  method  of  Case  I  should  always  be  applied  first. 

1.  Factor  2  a^if  —  8  axy^. 

We  have,  2  ax^y'^  —  8  axy^  =  2  axy'^(x^  —  4  ?/-),  by  Case  I. 
Whence  by  Case  IV, 

2 ax^y"^  -  8 axy^  =  2 axy'^ix  -h  2y)(x  -  2y),  Ans. 

If  the  given  expression  is  in  the  form  of  the  difference 
of  two  perfect  squares,  it  is  always  better  to  first  apply  the 
method  of  Case  IV. 

2.  Factor  a«-6«. 

We  have,      a^  -  b^  =  (a^  +  63)  (^^3  _  ^3)^  by  Case  IV. 
Whence  by  Case  VI, 

a^-b^=  (rt  +  b)  (a2  -'a6  +  b^)  (a  -  b)  (a^  +  ab  +  b^) ,  Ans. 

3.  Factor  ic^  —  i/. 

We  have,        x^  —  y^  =  (x^  +  y^)  (x*  —  y^),  by  Case  IV 
=  (x4  +  ?/4)(x2  +  ?/2)(a:2-?/2) 

=  (ic*  +  y*)  (a;2  +  y^)  (x  +  y)  (x-y),  Ans. 


MISCELLANEOUS  AND   REVIEW  EXAMPLES. 
106.    Factor  the  following : 

1.  35a*b^-^9Sa'b^-49a'b.  10.  4.  a'b' +  ^  a'b'. 

2.  25  aV- 816V.  11.  a^  +  15  a6  +  56  61 

3.  a^  +  11^  +  18.  12.  xhf  -  23  xy +  132. 

4.  a'bc-\-acH-abH-bcd^      13.  lOS  x' -  36 x"  +  3 x". 

5.  6x^-6x\  14.  64:  a^b- 121  a'b\ 

6.  A97n^  +  567nn  +  16n\       15.  x^-1. 

7.  o? -10a +  24..  16.  x^  +  o^y  +  xy"^  +  f. 

8.  ar'4-17a;2-38a;.  17.  a=^6^  -  3  aft^  _  180. 

9.  a?-(p^c)\  18.  2aj2  +  20a^-782/^ 


FACTORING.  79 

Id.  SO x' -DO 3^-^-65x^-20 x\    25.  27d'-6'kxf. 

20.  l-a«.  26.  32a^  +  2/i«. 

21.  16  x'  -  1.  27.  8  a^b-  12 a%''  4- 162  ah^. 

22.  64  a2^2  _  80  ahc  +  25  &.        28.  1-11  m/i  -  60  mV. 

23.  15ac+18arf-356c-42  M  29.  {x  -  yf-{m  -  n)\ 

24.  100  iB«  -  49 1^.  30.  (1  +  ny-  4  n\ 

31.  64  0.^2"- 56  a;yz*  + 72  ar^2/22^ 

32.  3a«62_3air  5O.  ^x? ■\-2of -\6z^ -^SOxy. 

33.  ?M*  -  81.  51.  343  m^  +  216  n\ 

34.  8  ary  +  125.  62.  (9  a^  +  4)"  -  144  a\ 
36.  {yii^-n)^^-l{yii^n)-\\\.     53.  (.x-^  -fa;-  9)^  -  9. 

36.  a^^  -  15  ahxy  -  54  ^y.  64.  {a^-2af+2 {o?-2a)  +  l. 

37.  25  ar*  +  HO  xy  +  121  /.  55.  a^¥  -f  aV  -  6V  -  x^- 

38.  4a«-8a^-2a^-f  4«l  66.  a;«-256. 

39.  (5a;-8y)2-(4.x-9^)2.  57.  36a2-462_  49c2 +  28  6c. 
40.5ar'4-5ar^.  58.  m'' -  62b. 

41.  (a2  +  9)=^  -  36  a\  69.  (a.-2-f3  a;)^  +4  (ar^+3  a;)  +4. 

42.  a;^  -  (a;  -h  2f.  60.  a«  -  7a«  -  8. 

43.  aV-46V-9a2d2-f 36^2^2  61.  27 a« -  1000 ftW 

44.  (a;2-5a;)2-2(ar^-5a;)-24.  62.  128  -  m^ 

45.  16 a;*  -  72  ar^^z'  +  81  y".         63.  2 a26c-2  hh-A  Wc^-2  h&. 

46.  a«  -  2  a^  -f  1.  64.  (a2-f7a)2+4(a2+7a)-96. 

47.  64-a;«.  66.  a;^«  +  2af +  1. 

48.  45  ar^+18  a;*+60  a.-3+24  3?.    66.  {:^  -  4)^  -(x-{-  2)\ 

49.  9  (wi-n)2-12  (m-w)4-4.    67.  (a^  -  6^  4.  c^)^  -  4a2c2. 

68.  Resolve  x^  —  7/  into  two  factors,  one  of  which  is  x-\-  y. 

69.  Resolve  a^  —  b^  into  two  factors,  one  of  which  is  a  —  b. 


80  ALGEBRA. 

70.  Resolve  x^  -\-  y^  into  two  factors  by  the  method  of  §  104. 

71.  Resolve  a^  +  2/^  into  three  factors  by  the  method  of 

§103. 

72.  Resolve  1  —  w?  into  two  factors  by  the  method  of  §  104. 

73.  Resolve   a^  —  1    into  three  factors  by  the  method  of 

§103. 

74.  Faxjtor  3  {m  -f  nf  -  2  (m^  -  n^). 

3(w  +  n)2  -  2(m2  -  ii^)  =  S{m  +  n)'^  -  2(m  +  ?i)(wi  -  n) 
=  (m  +  n)  [S(m  +  «)  -  2(m  -  n)] 

=  (m  4- w)(w  +  5w),  Ans. 

75.  Factor  (a  +  6)^  -  (a  -  bf. 
By  the  method  of  §  103,  we  have 

{a  +  by-ia-by 

=  l{a  +  b)-(a-  &)][(«  +  6)2  +  (a  +  b)(a  -  b)+{a  -  6)2] 
=  (^aj-b  -a  +  6)(a2  +  2a6  +  62  +  a2  -  62  +  a'^-  2ab  +  62) 
=  2  6(3a2  4-  62),  Ans. 

Factor  the  following : 

76.  (m  -  xf  4-  Sx-^.  84.  a:'-b'+x'-y'-^2ax-\-2by. 

77.  a^  -  (a  -  bf.  85.  (a?  -  m)'^  -x(x^-  m^. 

78.  5  (aj2  -  2/')  4-  4  (.t  -  2/)'.  86.  (x  +  2/)-^  -  (a?  -  y)^. 

79.  (a'  +  53)  -  2  a6  (a  +  b).  87.  a^°  -  1. 

80.  a^-\-b^-d'-d^+2ab-2cd.  SS.  x' -{- x' -  a^  -  1, 

81.  (x  +  1)^'  +{x-  ly.  89.  (a^  -  1)  -  (a  -  1)^ 

82.  (a^  +  f)-\-x(x  +  yf.  90.  (3m-  2f  +  (2 m  +  1)^. 

83.  a' -a'-  a'  +  1.  91.  (c^  -f-%y-^  yh\ 

92.  a2  +  2562_16c2-9d2_10a6-24cd 

93.  (l  +  a^)+2(l-a)(l  +  a)2. 


HIGHEST  COMMON   FACTOR.  81 


X.    HIGHEST  COMMON  FACTOR.   , 

107.  The  Degree  of  a  rational  and  integral  monomial 
(§  69)  is  the  number  of  letters  which  are  multiplied  to- 
gether to  form  its  literal  portion. 

Thus,  2  a  is  of  the  first  degree ;  5  a6  of  the  second  degree ; 
Sa^b^,  being  the  same  as  Saabbb,  is  of  the  fijlh  degree;  etc. 

The  degree  of  a  rational  and  integi-al  monomial  is  equal 
to  the  sum  of  the  exponents  of  the  letters  involved  in  it. 

Thus,  a*b(^  is  of  the  eighth  degree. 

108.  A-  polynomial  is  said  to  be'  rational  and  integral 
when  each  term  is  rational  and  integral ;  as  2a^b  —3c-\-  d^ 

The  degree  of  a  rational  and  integral  polynomial  is  the 
degree  of  its  term  of  highest  degree. 
Thus,  2  a^b  —  3  c  +  di^  is  of  the  third  degree. 

109.  A  Prime  Factor  of  an  expression  is  a  factor  which 
cannot  be  divided  without  a  remainder  by  any  expression 
except  itself  and  unity. 

Thus,  the  prime  factors  of  6a^(x^  -^  1)  are  2,  3,  a,a,x-\- 1, 
and  x  —  1. 

110.  The  Highest  Common  Factor  (H.  C.F.)  of  two  or 
more  expressions  is  the  product  of  all  their  common  prime 
factors. 

It  is  evident  from  this  definition  that  the  highest  common 
factor  of  two  or  more  expressions  is  the  expression  of  high- 
est degree  (§  108)  which  will  divide  each  of  them  without  a 
remainder. 

111.  Two  expressions  are  said  to  be  priine  to  each  other 
when  unity  is  their  highest  common  factor. 


82  ALGEBRA. 

112.  Eequired  the  H.  C.  F.  of  a'b'c",  a^W&,  and  a^hc\ 
Resolving  each  expression  into  its  prime  factors,  we  have 

a'^h'^i?  =  aaaabbccc, 
o?W&  _  aabbbccccc, 
and  a^bd^  =  aaabcccc. 

Here  the  common  prime  factors  are  a,  a,  b,  c,  c,  and  c. 

Whence,  the  H.  C.  F.  =  aabccc  =  a^b&. 

It  will  be  observed,  in  the  above  result,  that  the  exponent 
of  each  letter  is  the  lowest  exponent  with  which  it  occurs  in  any 
of  the  given  expressions. 

113.  In  determining  the  highest  common  factor  of  alge- 
braic expressions,  we  may  distinguish  two  cases. 

114.  Case  I.  When  the  expressions  are  monomials,  or 
'Dolynomials  which  can  be  readily  factored  by  inspection. 

1.   Find  the  H.  C.  F.  of  28  a'W,  42  ab%  and  98  a^Wd^. 
We  have,  28  a%^  =  2^x1  x  a'^b^ 

42  ab^c  =  2  X  3  X  7  X  ab^c, 
and  98  a%^d^  =  2xT^x  a%^cP. 

By  the  rule  of  §  112,  the  H.  C.  F.  =  2  x  7  x  ab^  =  14  ab^,  Ans. 

EXAMPLES. 

Find  the  highest  common  factor  of : 

2.  2a%5a^b\  ^  4.   45cr6^120aV. 

3.  20a^y,15xy'.  5.   lS2a^yz',  S4:a^fz. 

6.  16mV,  56  m%2,  88  mV. 

7.  36  a'bc%  72  a^b%  ISO  ab^c^ 

8.  126  aV,  21  a'xV,  147  a'x^z. 

9.  140  mVa;2,  175m^n%  105  mhis^. 
10.   117a^62c«,  104a*6V,  156a^6V. 


HIGHEST   COMMON  FACTOR.  §3 

11.  Find  the  H.  C.  F.  of 

5  x^y  -  45  y?y  and  10  y^y"  +  40  ^y''  -  210  xy".. 
We  have,         5  a^y  —  45  x^y  =  5  yhf  {y?-  -  9) 

=  6x2y(x  +  3)(x-3),  (§99J 

and   10  xhP'  +  40  xV  _  2IO  xy"^  =  10  x^a  (x2  +  4  x  -21) 

=  2  X  5  X  X2/2  (x  +  7)  (X  -  3).   (§  100) 
By  the  rule  of  §  112,  the  H.  C.  F.  is  5xy  (x  -  3),  Ans. 

12.  Find  the  H.  C.  F.  of 

4 a^  —  4 a  + 1,  8 a^  —  1,  and  2am  —  m —  2a]i-\-  n. 
We  have,  4rt2  _  4^  _|.  1  =  (2a  _  i)2,  (§98) 

8a3- 1  =  (2a-l)(4a2  +  2a+l),     (§103) 
and  2  am  —  »n  —  2  an  +  n  =  (2  a—  1)  {m  —  71).  (§  93) 

By  the  rule  of  §  1 12,  the  H.  C.  F.  is  2  a  -  1,  Ans. 

Find  the  highest  common  factor  of : 

13.  6  a^b^  -  15  a^b\  12  a'h  +  21  a^h\ 

14.  68  (m  +  nf  (m  -  n)\  85  (m  +  nf  (m  -  n). 

15.  ar^-9/,  a.-2-6a^  +  9/. 

^     16.  3a3-21a2_a  +  7,  a2  +  6a-91. 

^    17.  2a8a;  +  4aV  +  2aar8,  3a^x  +  3aa;^ 

18.  m«  -  27,  m^  -  11  m  +  24. 

19.  ac-^ad  —  bc  —  bd,  a^  —  6ab-\-5  b\ 

20.  a;  +  4a^4-4a^,  4  +  44a;4-72iB2. 

21.  80n'-5n«,  20ri^4-5  7i2. 

^      22.  a2  +  62_^_j..2a6^  a2-62_c2  4-26c.     ^ 

23.  ar^H-2a;-24,y-14a;-^40,  ar^-8a;-f  16. 

24.  9,a2  _  12  a  -f-  4,  9  a^  -  4,  18  a^  -  12  a\ 

25.  iB2-6a;-27,  a.'24.6a;4-9,  3^^21. 


84  ALGEBRA. 

26.  a^  -f  13  cr  +  40  a,  a^  -  a^  -  30  a',  a' -\- 2  a'  -  15  a^ 

27.  m^  -  4  m,  m^  +  9  m^  -  22  m,  2  m^  -  4 1??;^  -  3  m^  +  6  m. 

28.  x'^-^f,  x'-4.y',  x'-9xy-^Uy\ 

29.  Sa'-a'b-^-Sab-  b%  27  a^  -  6^  9  a^  _  6  a?>  +  61 

30.  27a.'3  +  125,  9a?2-25,  9  a;^  ^  39  3.  _^  25. 

31.  ^y-:f?f-2^xf,  2 x'y'' ^-22 o^f ^m xy\  Zx'^y-^^:f?f. 

32.  16  m^  —  wS  16  m''  —  8  mV  +  ri^  2  ma;  +  2  m?/  —  nic  —  wi/. 

33.  (X'  —  ^,  a^  —  a^x  —  aa^  +  .t^,  3  a^  —  3  a^a;  4-  5  ax^  —  5  x^. 

115.  Case  II.  IF/ieii  the  expressions  are  polynomials  which 
cannot  be  readily  factored  by  inspection. 

The  rule  in  Arithmetic  for  the  H.  C.  F.  of  two  numbers 
is: 

Divide  the  greater  number  by  the  less. 

If  there  be  a  remainder,  divide  the  divisor  by  it;  and  con- 
tinue thus  to  make  the  remainder  the  divisor,  and  the  preceding 
divisor  the  dividend,  until  there  is  no  remainder. 

The  last  divisor  is  the  H.  C.  F.  required. 

Thus,  let  it  be  required  to  find  the  H.  C.  F.  of  169  and 

546. 

169)546(3 

507 

~39)169(4 
156 

~13)39(3 
39 

Then,  13  is  the  H.  C.  F.  required. 

116.  We  will  now  prove  that  a  rule  similar  to  that  of 
§  115  holds  for  the  H.  C.  F.  of  two  algebraic  expressions. 

Let  A  and  B  be  two  polynomials,  the  degree  of  A  (§  108) 
being  not  lower  than  that  of  B. 


HIGHEST   COMMON   FACTOR.  85 

Suppose  that  B  is  contained  in  A  p  times,  with  a  remain- 
der (7;  that  C  is  contained  in  B  q  times,  with  a  remainder 
D ;  and  that  D  is  contained  in  C  r  times,  with  no  remainder. 

To  prove  that  D  is  the  H.  C.  F.  of  A  and  B. 

The  operation  of  division  is  shown  as  follows : 

B)A{p 
pB 

~C)B(q 
qC 

D)C{r 
rD 

0 
We  will  first  prove  that  Z)  is  a  common  factor  of  A  and  B. 
Since  the  minuend  is  equal  to  the  subtrahend  plus  the 
remainder  (§  35),  we  have 

A=pB+C,  (1) 

B=:qC-\-D,  (2) 

and  C  =  rD. 

Substituting  the  value  of  G  in  (2),  we  obtain 

B  =  qrD  +  D  =  D(qr  -\-  1).  (3) 

Substituting  the  values  of  B  and  C  in  (1),  we  have 

A=pD  (qr  -{-l)-^rD  =  D  (pqr  -\-p  +  r).  (4) 

From  (3)  and  (4),  Z>  is  a  common  factor  of  A  and  B. 
We  will  next  prove  that  every  common  factor  of  A  and  B 
is  a  factor  of  D. 

Let  F  be  any  common  factor  of  A  and  B ;  and  let 

^  =  mi^  and  ^  =  nF. 
From  the  operation  of  division,  we  have 

C=A-pB,  (5) 

and  D=B-  qC.  (6) 


86  ALGEBRA. 

Substituting  the  values  of  A  and  B  in  (5),  we  have 

C=  mF  —  pnF. 
Substituting  the  values  of  B  and  C  in  (6),  we  have    . 

D  =  nF  —  q  (mF  —  piiF)  =  F(n  —  qm-{-  jyqn). 
Whence,  i^  is  a  factor  of  D. 

Then,  since  every  common  factor  of  A  and  5  is  a  factor 
of  D,  and  since  D  is  itself  a  common  factor  of  A  and  B,  it 
follows  that  D  is  the  highest  common  factor  of  A  and  B. 

117.  Hence,  to  find  the  H.  C.  F.  of  two  polynomials, 
A  and  B,  of  which  the  degree  of  A  is  not  lower  than  that 
ofB, 

Divide  Ahy  B. 

If  there  he  a  remainder,  divide  the  divisor  by  it;  and  con- 
tinue thus  to  make  the  remainder  the  divisor,  and  the  2^receding 
divisor  the  dividend,  until  there  is  no  remainder. 
The  last  divisor  is  the  H.  C.  F.  required- 
Note  1.     Each  division  should  be  continued  until  the  remaiudei- 
is  of  a  lower  degree  than  the  divisor. 

Note  2.  It  is  of  the  greatest  importance  to  arrange  the  given 
polynomials  in  the  same  order  of  powers  of  some  common  letter 
(§  33),  and  also  to  arrange  each  remainder  in  the  same  order. 

1.   Find  the  H.  C.  F.  of 

6  a;2  _  13  a;  _  5  and  18  a^  -  51  a^  +  13  a;  +  5. 

6 ic2  -  13a;  -  5)18  a;3  -  51  a;2  +  13 a;  +  5(3 a;  _  2 
18x3 -39x2 -15  a; 


-  12  x2  +  28  X 

-  12  x2  +  26  X  +  10 


2x-    5)6x2 -13x-5(3x  +  l 
6x2 -15x 


2x 

2x-5 
Whence,  2x  —  5  is  the  H.  C.F.  required. 


HIGHEST   COMMON   FACTOR.  87 

Note  3.  If  the  terms  of  one  of  the  given  expressions  have  a 
common  factor  which  is  not  a  common  factor  of  the  terms  of  the  other, 
it  may  be  removed ;  for  it  can  evidently  form  no  part  of  the  highest 
common  factor.  In  like  manner,  we  may  divide  any  remainder  by  a 
factor  which  is  not  a  factor  of  the  preceding  divisor. 

2.   Find  the  H.  C.  F.  of 

6a^  -25x^-^Ux  and  6ax^-{-llax-  10  a. 

In  accordance  with  Note  3,  we  remove  the  factor  x  from  the  first 
expression,  and  the  factor  a  from  the  second. 

6x2  -  25x  -f-  14)6x2  +  llx  -  10(1 
6x2-25x+14 


36x- 

24 

divide  this  remainder  by 

3x  -2)6x2- 

6x2- 

12  (Note  3). 
-25x  +  14(2x 

-  4x 
-21x 

-  i!l  X  +  14    . 

;-7 

Whence,  3  x  -  2  is  the  H.  C.  F.  required. 

Note  4.     If  the  given  expressions  have  a  common  factor  \yhich 
can  be  seen  by  inspection,  remove  it,  and  find  the  H.  C.  F.  of  the ' 
resulting  expressions.     The  result,  multiplied  by  the  common  factor, 
will  be  the  H.  C.  F.  of  the  given  expressions. 

3.   Find  the  H.  C.  F.  of 

2a^-Sa^b-2  ab^  and  2  a»  -f-  7  a^ft  +  3  ab\ 

In  accordance  with  Note  4,  we  remove  the  common  factor  a,  and 
find  the  H.  C.F.  of  2a2  -Sab  -  2  62  and  2  a2  +  7  ab  +  362. 

2a2-3a6-262)2a2+    lab-{-Sh^(l 
2a2-    3ab-262 
5  6)10a6  +  5  62 
2a  +  b 
2  a  +  6)2  a2  -  3  a6  -  2  62(a  -  2  6 
2  a2  4-     ab 

-4ab 

-4  ah  -2  ir^ 


Multiplying  2  a  +  6  by  a,  the  required  II.  C.  F.  is  «(2  a  +  6). 


88  ALGEBRA. 

Note  5.  If  the  first  term  of  the  dividend,  or  of  any  remainder,  is 
not  divisible  by  tlie  first  term  of  the  divisor,  it  may  be  made  so  by 
multiplying  the  dividend  or  remainder  by  any  term  which  is  not  a. 
factor  of  the  divisor. 

Note  6.  If  the  first  term  of  any  remainder  is  negative,  the  sign 
of  each  term  of  the  remainder  may  be  changed. 

4.   Find  the  H.  C.  F.  of 

2a^-3a;2^2aj-8and3a^-7a^  +  4a;-4. 

3x3-    7x=^  +  4a;-4 
2 


2x^-Sx^-\-2x~S)6x^-Ux^-\-Sx-    8(3 
6x3-    9x2  4-6x-24 

-    5x-^  +  2x  +  16 

2x3-3x2+      2x-      8 
5 


6x2 -2x- 16)10x3- 15x2+    lOx-    40(2x 
10x3-    4x2-    32 X 


-11x2+    42  X-    40 
'    '  5 

-  55x2  + 210 x-200(-  11 
-55x2+    22X+176 

188)  188  x  -  376 

X-      2 

X- 2)5x2-    2x-16(5x  +  8 
5x2 -lOx 
8x 
8x-16 

Whence,  x  —  2  is  the  H.  C.  F.  required. 

In  the  above  example,  we  multiply  3  x3  —  7  x2  +  4  x  —  4  by  2  in 
order  to  make  its  first  term  divisible  by  2  x3. 

We  change  the  sign  of  each  term  of  the  first  remainder  (Note  6), 
and  multiply  2  x3  —  3  x2  +  2  x  —  8  by  5  to  make  its  first  term  divisible 
by  5x2. 

We  multiply  the  remainder  —  1 1  x2  +  42  x  —  40  by  5  to  make  its 
first  term  divisible  by  5  x2. 


HIGHEST  COMMON  FACTOR.  89 

EXAMPLES. 
Find  the  H.  C.  F.  of : 

6.  2  a^  +  7  a  +  6,  6  «-  +  H  a  +  3. 

7.  4x2  +  13a;  +  10,  Gx-^H-Sa^-U. 

8.  ar^  +  5a;-24,  a^  +  4a!2-26a;-f  15. 

9.  3m2  +  m-2,  4m3  +  2m2_mH-l. 

10.  lSa^-^9ab-ob-,  24:a'-29ab-{-7b'. 

11.  12  a^  —  5  a^x  —  11  aor  +  6  af^,  15  a*^  +  11  arx  —  8  aa^  —  4  ic^. 

12.  4a;«-12ar'4-5a;,  2a;^  +  ur^-7x'2-20a;. 

13.  S3^-{-13x'y-\-12xy',  9a^y-22xf-Sy\ 

14.  4a*-lla2  +  5a  +  12,  6a'^-lla^  +  13a'^-4a2. 

15.  2  m^-f  5  m^M— 2  ??iV-f  3  7nn%  6  m^7i— 7  ??t^;A^+5  mn^— 2  ^^^ 

16.  3ar'-4aj-4,  3a;*  -  7a^4-6ar^-9a;  + 2. 

17.  3a*  +  5a3  +  12a2-f  8,  6a*  +  lOa--^  +  19a- ^  10a -4. 

18.  2m^-3m^x-Smx^-3a^, 

3  m*  —  7  w?x  —  5  m  V  —  mar^  —  6  a;"*. 

19.  2a*-a»-4a2  +  3a,  4a*-6a='  +  a-  +  4a-3. 

20.  m^  +  8  m2,  w^•^  -  2  »i*  -  15  ^u^  -  14  m\ 

21.  4a*-22a=^64-6a2/r'+20a6^  9 a''b-^2  tv^b^-l^aif+l^b^ 

22.  4ar'  +  9x--9,  2.r*  + 11  ar'^  + 14.i-- 5a;- 6. 

23.  3a*-6a^+4a=^+4a-4,  3  a^+ 3  a*- 11  a^- 2  a^  +  6  a. 

24.  a«  +  2a2-2a  +  24,  a*  +  2a3  -  11  a^  -  6a  +  24. 

25.  2x^-3x''y^-3x'y''-3xf  +  y', 

2  a;*  +  ar^y  —  3  a;^2/^H"  5  a;^  —  2  ?/*. 

26.  2a;*  +  a;3-9a;2_^a;4-l,  2a;*  -  9a.'^4- 12a;2  _  3^_2. 

27.  2y?-7x^  +  7  x-2,  x^-3x''-^nx'-4.x  +  4t. 

28.  a«  a;  -  aV  -  a?:t?  -  aV  -  2  ax^, 

a^x  +  3  a*ar^  —  a^a^  —  4  a  V  —  oar*. 


90  ALGEBRA. 

118.  The  H.  C.  F.  of  three  expressions  may  be  found  as 
follows : 

Let  A,  B,  and  C  be  the  expressions. 

Let  G  be  the  H.  C.  F.  of  ^  and  B\  then,  every  common 
factor  of  G  and  (7  is  a  common  factor  of  A,  B,  and  C. 

But  since  every  common  factor  of  two  expressions  exactly 
divides  their  highest  common  factor  (§  116),  every  common 
factor  of  A,  B,  and  C  is  also  a  common  factor  of  G  and  C. 

Whence,  the  highest  common  factor  of  G  and  C  is  the 
highest  common  factor  of  A,  B,  and  C. 

Hence,  to  find  the  H.  C.  F.  of  three  expressions,  find  the 
H.  C.  F.  of  two  of  them,  and  theii  of  this  result  and  the  third 
expression. 

We  proceed  in  a  similar  manner  to  find  the  H.  C.  F.  of 
any  number  otf  expressions. 

1.   Find  the  H.  C.  F.  of 
iK3_7a;4-6,  ;x?-{-Zx^-Ux-^12,  and  o?-Bx^  +  lx-^. 
TheH.C.F.  of  ic^  -  7  a:  +  6  and  x^  +  3:^2-  16x  +  12  isx2-3x  +  2. 
The  H.  C.  F.  of  x^  -  3  x  +  2  and  a;^  -  5  x'^  +  7  x  -  3  is  x  -  1,  Ans. 

EXAMPLES. 
Find  the  H.  C.  F.  of: 

2.  ^x'-llx^-S^,  4.x^-12x-27,  6a^-31a;+18. 

3.  Sa^  +  22a-{-5,  12a^-lSa-4:,  20  a' +  29  a -\- 6. 

4.  15m' -4.771 -32,  18m2  +  3m-28,  21m' +  25  m -A. 

5.  5a'-\-23ab-10b',  5 a^ -\- S3 a'b -\- ^6 ab' -  24: b% 

5a^-{-  SSa'b  +  59  ab'  -  30  61 

6.  a^-{-x^-14:X-24,   x^  -  3x^ -6x-{-8,   a^ -\-4:X^ -i- x  -  6. 

7.  a'-a'-5a-3,  a'  +  2a'-a-2,  a^  -  2a' -2a-\-l. 

8.  2m^  +  97n'-6m-5,  3m^  +  10m'-23m-\-10, 

6m'-7m'-m-\-2. 

9.  2x^-x'y-27xf  +  36f,  2  a^  -  5  x'y  -  37  xif -\- 60  f, 

2x^-  19a^^  +  51xf  -  4.5  f. 


LOWEST  COMMOX  MULTIPLE.  91 


XI.  LOWEST  COMMON  MULTIPLE, 

119.  A  Common  Multiple  of  two  or  more  expressions  is 
an  expression  which  can  be  divided  by^each  of  them  with- 
out a  remainder. 

120.  The  Lowest  Common  Multiple  (L.  C.  M.)  of  two  or 
more  expressions  is  the  product  of  all  their  different  prime 
factors  (§  109),  each  taken  the  greatest  number  of  times 
that  it  occurs  as  a  factor  in  any  one  of  the  expressions. 

121.  Required  the  L.  C.  M.  of  a'b^c,  ah'd\  and  6Vd*. 
Here,  the  different  prime  factors  are  a,  h,  c,  and  d;  a 

occurs  twice  as  a  factor  in  a?bh ;  h  five  times  as  a  factor  in 
a6'd^;  c  three  times  as  a  factor  in  6Vd*;  and  d  four  times  as 
a  factor  in  h^&d*. 

Whence,  the  required  L.  C.  M.  is  a^ftVd^  (§  120). 

It  will  be  observed,  in  the  above  result,  that  the  exponent 
of  each  letter  is  the  highest  exponent  icith  which  it  occurs  in 
any  one  of  the  given  expressions. 

122.  It  is  evident  from  the  definition  of  §  120  that  the 
lowest  common  multiple  of  two  or  more  expressions  is  the 
expression  of  loivest  degree  (§  108)  which  can  be  divided  by 
each  of  them  without  a  remainder. 

123.  If  two  expressions  are  prime  to  each  other  (§  111), 
their  product  is  their  lowest  common  multiple. 

124.  In  determining  the  lowest  common  multiple  of 
algebraic  expressions,  we  may  distinguish  two  cases. 

125.  Case  I.  When  the  expressions  are  monomials,  or 
polynomials  which  can  he  readily  factored  by  inspection. 


92  ALGEBRA. 

1.  Find  the  L.  C.  M.  of  2Sa''h%  Uh(?,  and  63 cU 
We  have,  28  a^h^  =  2^  x  7  x  a^ft^, 

54  6c3  =  2  X  33  X  6c3, 
and  63  cH  =  32  x  7  x  cH. 

By  the  rule  of  §  121,  the  L.  C.  M.  =  2^  x  3^  x  7  x  a^h'^cH 

=  756  a'^b^c%  Ans. 

EXAMPLES. 
Find  the  lowest  common  multiple  of : 

2.  5ab%  7a^b\  6.   55  xy,  70  yz,  77  zx. 

3.  12  xy%  54.  yz\      ■  7.   50  a'b%  60a'b%  75a'b\ 

4.  24  m^  4.5  n\  8.    Wx'f,  21  fz,  33  a^;?^. 

5.  72  a%,  96  6V.  9.   20  a6^  27  6V,  OOc^cZ^. 

10.  36m^nic,  40mwy,  48  7iV?/. 

11.  56a26c«,  84a36«d^  126  aVt^^^ 

12.  Find  the  L.  C.  M.  of  x'-^-x-Q,  a^-4a;  +  4,  and 
x^  —  9x. 

We  have  x"^  -h  x  -  6  =(x  +  S)(x  -  2),  (§  100) 

aj2_4a;  +  4=(x-2)2,  (§98) 

and  x^-9x  =  x(x  +  S)(x-S).  (§99) 

By  the  rule  of  §  121,  the  L.  C.  M.  =  x(x  -  2)2(a;  +  3)  (x  -  3),  Ans. 

Find  the  lowest  common  multiple  of : 

13.  a"  -  62,  a^  _|_  2  a6  +  61 

14.  m^  +  '^n,  mn  —  ti^. 

15.  a^-9,  ar^  +  lOaj  +  21. 

16.  a;^-18a^  +  81aj^  x'3-13a;2H-36a;. 

17.  a^  -  3a6  +  2  6^  ac  +  ad  -be-  bd. 

18.  a2-h2aa;  +  a^,  a^  +  a^. 

19.  l-8ar^,  l  +  9a;-22a;l 


LOWEST  COMMON  MULTIPLE. 


93 


20.  m^  4- 13  m^n  +  40  mn^,  mhi  —  mii^  —  30  n^. 

21.  4a;2_25^  2aj3-5a^-4a;  +  10. 

22.  x"  -^^ax'-lS  a%  ax"  +  15  a-x  +  54  a\ 

23.  4  a^  -  2  «6,  4  a6  +  2  6^,  4  a^  -  h\ 

24.  6a:2_^-^Q3^^  9a,'2/-15/,  ^Qs^y-imxif. 

25.  4m2-8i^  +  4,  6/?i2  +  12?yi  +  6,  m^  -  1. 

26.  a2  -  12  a  +  35,  a^  +  2  a  -  63,  a^  -  3  a  -  108. 

27.  a;*  —  4  aa^  +  4  aV,  a^  +  4  aa;  H-  4  a^,  oa;^  —  4  a?x. 

28.  3a:2_g,p_72,  4ar^4-8x-192,  2a^-24a;  +  72. 

29.  x^y-xi^,  a^-ff  x^-2xy  +  y^. 


■^  if 


30.   ar^  +  2/2  _  ^2  _  2a^,  a^  _  y-'  _  ^2 


^2;. 


^ 


31.  16  m^  -  9  ?i^  8  ahhn  -  6  aft^^i,  16  m^  -  24  m^i  +  9  n\ 

32.  a-^-a,  a^-9a^-10a,  a*  -  a^ -\- a^  -  a. 

33.  aj2  +  4.ri/  +  4/,  a.'^-h  a;?/  -  2/,  ar^  +  82/^. 

34.  2a^-2a'--ia,  3a*-6a^-9a%  4 a^ -f  20 a^ -f  16 a^. 

35.  27a;«-8,  9a.'2-4,  9ar'-12a;  +  4. 

36.  4ar^-4m2,  ex-{-6m,  Sx^-^-Sm^  9a;-9m. 

37.  x*-y\  x^-\-2x'fVy*,  x'-2a^y^  +  y\ 

38.  a^  4-  b%  a^  -  b%  (a"  +  ft?)^  -  ci'b'. 

39.  a2- 11  aa;  + 18^2,  a^- 5aa;  -  Ua:^^  a*- 8aV+ 16  a;*. 

40.  m^  —  71^,  m^  —  m^n  —  mn^  +  n^,  m^  +  m^^i  —  mn^  —  n^. 

41.  a2+62_c2+2a6,  a^- 62_  c2_2  6c,  a^- 6^+ c^- 2ac. 


126.   Case    II.    When    the   expressions    are   polynomials 
which  cannot  be  readily  factored  by  inspection. 

Let  A  and  B  be  any  two  expressions. 
Let  F  be  their  H.  C.  F.,  and  M  their  L.  C.  M, ;  and  sup- 
pose that  A  =  aF,  and  B  =  bF. 


94  ALGEBRA. 

Then,  Ax  B==  ahF\  (1) 

Since  F  is  the  H.  C.  F.  of  A  and  B,  a  and  b  have  no  com- 
mon factors ;  whence,  the  L.  C.  M.  of  aF  and  bF  is  abF. 

That  is,  M=abF. 

Multiplying  each  of  these  equals  by  F,  we  have 

FxM=  abF\  (2) 

From  (1)  and  (2),      AxB  =  FxM.  (§  9,  4) 

That  is,  the  product  of  two  expressions  is  equal  to  the  prod- 
uct of  their  H.  C.  F.  and  L.  C.  M. 

Therefore,  to  find  the  L.  C.  M.  of  two  expressions, 
Divide  their  product  by  their  highest  common  factor ;  or, 
Divide  one  of  the  expressions  by  their  highest  common  fac- 
tor, and  multiply  the  quotient  by  the  other  expression. 

'  1.   Find  the  L.  C.  M.  of 

6x^-17 aj  +  12  and  12a^-4a;-21. 

6x2- 17 x  + 12)12x2-    4x-21(2 
12x2-  84x  +  24 


15)30  X- 45 

2x-   3)6x2  -  17  X -f  12(3  X- 
6x2-    9x 

-4 

-    8x 

-    8x4-12 

Then  the  H.  C.  F.  of  the  expressions  is  2  x  -  3. 

Dividing  6 x2  -  17 x  +  12  by  2 x  -  3,  the  quotient  is  3x  -  4. 

Whence,  the  L.  C.  M.  =(3x  -  4) (12  x2  -  4x  -  21),  Ans. 

EXAMPLES. 
Find  the  L.  C.  M.  of: 

2.  2aj2-3ic-35,  2a;2-19a;  +  45. 

3.  3a2-13a  +  4,3a2-f  14a-5. 

4.  6a2  +  25a6  +  2462,  12a2  +  16a6-362. 

5.  &:»?^-llx'y-2xy\^x'y  +  21xy''-\-10f. 


LOWEST  COMMON  MULTIPLE.  95 

6.  12m2-21m-45,4m3-llm2-6m4-a 

7.  2a^-5a^-lSa-9,3a^-Ua'-a-\-6. 

8.  2a^x-\-a^x^-[-2aa^-hSx\  2  a^x -\- 5  a'sc^ -\- 2  as^  -  x\ 

9.  2  a^  -  5  a6  +  3  b%  a*  -\- a^b  -  5  a'b''  +  2ab^-\-  b\ 

10.  6a^-7ic2  +  5a;-2,4a^-5ar^  +  4a;-3. 

11.  2a-^-5a2  +  a  +  2,4a3-9a-4. 

12.  3  m^  —  7  m^n  +  4  mn^,  6  mhi  —  4  m^7i^  —  14  mn^  —  4  n*. 

13.  a'  +  2a*-5a^-hl2a%Sa^-\-lla'-6a*-7a'-{-4a\ 

14.  3a^-2a.-3-12a^-a;  +  6,  3a;* +  7353  +  60^2  _  2a;  _  4. 

127.  The  L.  C.  M.  of  three  expressions  may  be  found  as 
follows : 

Let  A,  B,  and  C  be  the  expressions. 

Let  M  be  the  L.  C.  M.  of  A  and  B ;  then,  every  common 
multiple  of  M  and  C  is  a  common  multiple  of  A,  B,  and  C. 

But  since  every  common  multiple  of  two  expressions  is 
exactly  divisible  by  their  lowest  common  multiple,  every 
common  multiple  of  A,  B,  and  C  is  also  a  common  multiple 
of  M  and  C. 

Whence,  the  lowest  common  multiple  of  M  and  C  is  the 
lowest  common  multiple  of  A,  B,  and  C. 

Hence,  to  find  the  L.  C.  M.  of  three  expressions,  find  the 
L.  C.  M.  of  two  of  them,  and  then  of  this  result  and  the  third 
expression. 

We  proceed  in  a  similar  manner  to  find  the  L.  C.  M.  of 
any  number  of  expressions. 

EXAMPLES. 
Find  the  L.  C.  M.  of : 

•^    1.   2a^  +  a;-15,  2a!-  +  7a:  4-3,  2a;2  +  9a;  +  9. 

2.  3a2  +  a-2,6a-  +  ll(H-5,9a2  +  5a-4. 

3.  2m2-5m4-2,  3 wi" -  10 ?n,  +  8,  4 m^  +  10 m  -  6. 

4.  2a^-  5.^  ^  3a;,  4a;*  -  11  a^-Sop',ex^-  x^  -2x\ 

.     5.   a3-2a2-5a  +  6,a3-3a2-a  +  3,a3-|-4a2  +  a-6. 


96  ALGEBRA. 


XII.    FRACTIONS. 

128.  The  quotient  of  a  divided  by  b  is  Avritten  -  (§  3). 

The  expression  -  is  called  a  Fraction ;  the  dividend  a  is 
b 

called  the  numerator,  and  the  divisor  b  the  denominator. 

The  numerator  and  denominator  are  called  the  te7'ms  of 
the  fraction. 

129.  Let  ^  =  x.  (1) 

Then  since  the  dividend  is  the  product  of  the  divisor  and 
quotient  (§  54),  we  have 

a  =  bx. 

Multiplying  each  of  these  equals  by  c  (§  9,  1), 

ac  =  bcx. 

Regarding  ac  as  the  dividend,  be  as  the  divisor,  and  x  as 
the  quotient,  this  may  be  written 

Tc  =  -  (^) 

From  (1)  and  (2),  ^  =  ^.  (§9,4) 

be      b 

That  is,  if  the  terms  of  a  fraction  be  both  multiplied,  or  both 
divided,  by  the  same  expression,  the  value  of  the  fraction  is  ^lot 
altered. 

130.  By  the  Law  of  Signs  in  Division  (§  55), 

■j-a _  —  a _      -]- a _      —a 

That  is,  if  the  signs  of  both  terms  of  a  fraction  be  changed, 
the  sign  before  the  fraction  is  not  changed  ;  but  if  the  sign  of 
either  one  be  changed,  the  sign  before  the  fraction  is  changed. 


FRACTIONS.  97 

If  either  term  is  a  polynomial,  care  must  be*  taken,  on 
'.^hanging  its  sign,  to  change  the  sign  of  eoc/i  of  its  terms. 

Thus,  the  fraction      ~    ,  by  changing  the  signs  of  both 
c  —  d  ,  _ 

numerator  and  denominator,  can  be  written  (§  41). 

d  —  c 

131.   It  follows  from  §§49  and  130  that 

If  either  term  of  a  fraction  is  the  indicated  product  of  two 
or  more  expressions,  the  signs  of  any  even  number  of  them 
may  he  changed  without  changing  the  sign  before  the  fraction  ; 
but  if  the  signs  of  any  odd  number  of  them  be  changed,  the 
sign  before  tJie  fraction  is  changed. 

Thus,  the  fraction ^^^ may  be  written 

a  -^b  b  —  a  b  —  a  , 

• ,  etc. 


{d  -  c)  (/-  e)'  (d  -  c)  {e  -/)'       {d  -  c)  (/-  e)' 


REDUCTION  OF  FRACTIONS. 

132.  To  Reduce  a  Fraction  to  its  Lowest  Terms. 

A  fraction  is  said  to  be  in  its  lowest  terms  when  its  numer- 
ator and  denominator  are  prime  to  each  other  (§  111). 

133.  Case  I.    When  the  mimerator  and  denominator  can 

be  readily  factored  by  inspection. 

By  §  129,  dividing  both  terms  of  a  fraction  by  the  same 
expression,  or  cancelling  common  factors  in  the  numerator 
and  denominator,  does  not  alter  the  value  of  the  fraction. 

We  then  have  the  following  rule : 

Resolve  both  numerator  and  denominator  into  their  factors, 
and  cancel  all  that  are  common  to  both. 

24  o^b^c 

1.   Reduce  — —  to  its  lowest  terms. 

^Od'bH 

We  have  24  gSftgc  ^  28  x  3  x  gSft^c 

'  40  a'^hhl     28  X  5  X  a%'^d 


^Uh 


98  ALGEBRA. 

Cancelling  the  common  factor  2^  x  a^h'^,  we  obtain 
24  a%'^c      3  ac 


40  o^hH      5  d 


Ans. 


a^  —  27 

2.   Reduce  — to  its  lowest  terms. 

nr  —  2x  —  'd 

We  have,         _^slIL.  =  (x  -  S)(x^  ^  Sx  +  9)  ^^^ 

x+1 
Note.    If  all  the  factors  of  the  numerator  be  cancelled,  unity  re- 
mains  to  form  a  numerator ;  thus,  -^-^  = 

If  all  the  factors  of  the  denominator  be  cancelled,  the  division  is 
exact. 

EXAMPLES. 

Reduce  each,  of  the  following  to  its  lowest  terms : 

3    ^^.  6     ^^^^'  9       ^^^^^ 


a6V  12a%'  lOSa^ftV 

7m%«p  -    56a%V  -^    60 


54a;y  120a.Vg^  -.     126a^6V 

45^*  ISit-V  ■  *     98a6V* 

j2    3a^6-6a^6^  ^y    m^-m^-SGrn 


4a262  -  8a63  m^  +  m^  -  42m2 

^g      6a?^i/  +  8ar^y^  ^g^  0^  +  2/^ 


15i»y  +  20a^/  '   2a^y-2x'y^  +  2xf 

j^    a^  +  7a  +  10  jg    64a^  +  112 a^a;  + 49 ga;^ 
a^  -f-  4  a  —  5  64  a^cc  —  49  ic^ 

jg    a^-8a^  +  12a;  gQ    a^-Umx  +  ABm^ 
a^-12a;  +  36*  '   of  -  2  7nx -15m^' 

jg    25a^  +  20a6  +  4&^  ^^  a^-S 


25a2-462  a«_2a2  +  a-2 


FRACTIONS.  99 


6?/i-^  +  8m2-9m-12  ((i'-\-6a-\-d)(a--a-6) 

a^-/  +  g^  +  2a^  og    (a  +  6)^-(c  +  cZ)^ 

'*''•    x'-y^-z^-\-2yz  '    (a -rf)--(6  -  c)^ 

04  27a-^  +  646-^  2^    12ar^  + 8ar^ -3a; -2 

9a2-f-24a6  +  1662*  *    18af^- 9a;2  _  g^.^  4* 

28.  Reduce  j^ ^ to  its  lowest  terms. 

}r  —  (V- 

___   ,  ax-hx-ay  -\-  by     (a  -  b)  (x  -y)  ...  -  ^   ._. 

Changing  the  signs  of  the  factors  of  the  numerator  (§  131),  we  have 

ax  -  hx  —  ay  -\-  by  _{b  -  d){y  -  x)  _y  -  X 
62 -a2  ~  (6  +  a)(&  -  a)  ~  6  +  a' 

Reduce  eax;h  of  the  following  to  its  lowest  terms : 

29.  „'^-'"'  .  ■  32. 


Ans. 


30. 


7^2 -7m  4- 12 

14ar^-4ar^ 
4ar'-28a;  +  49* 


2ac-26c-od-|-6d 

d'-4.<^ 

1- 

-lla  +  18a2 

8a3-l 

a?- 

-(b-^cy 

134.  Case  II.  When  the  numerator  and  denominator  can- 
not he  readily  factored  by  inspection. 

Since  the  H.  C.  F.  of  two  expressions  is  the  product  of  all 
their  common  prime  factors  (§  110),  we  have  the  following 
rule: 

Divide  both  numerator  and  denominator  by  their  highest 

common  factor. 

2  a^  —  5  a  4-  3 

1.   Reduce ^^-—  to  its  lowest  terms. 

■6a2_a_l2 

By  the  rule  of  §  117,  we  find  the  H.  C.  F.  of  2a2-5a  +  3  and 
6  a2  -  a  -  12  to  be  2  a  -  3. 


100  ALGEBRA. 

Dividing  2  a^  -  5  a  +  3  by  2  a  -  3,  tlie  quotient  is  a  -  1. 
Dividing  6  a^  —  a  —  12  by  2  a  —  3,  the  quotient  is  3  a  +  4. 

Whence,  2a^-6a  +  S  ^  a-1     ^^^ 

6a2_a_i2      3a +  4' 


EXAMPLES. 
Reduce  each  of  the  following  to  its  lowest  terms : 

•  5a;2-23a;-42* 

3    2a^  +  ft-10  g 

•  4a2  +  8a-5* 

.      2oiy^  —  xy  —  15  y^  q 

2x'-15xy  +  2'7y^' 

f.    6  m^  —  13m  +  6  ^q 

•  9m2  +  6m-8* 

6  a.-^  +  3a.--10  ^^ 

aj3  +  2a.-2-14a;  +  5* 

135.  To  Reduce  a  Fraction  to  an  Integral  or  Mixed  Ex- 
pression. 

An  Integral  Expression  is  an  expression  which  has  no 
fractional  part ;  as  2  xy,  or  a  +  6. 

An  integral  expression  may  be  considered  as  a  fraction 
whose  denominator  is  1 }  thus,  a  +  6  is  the  same  as  -^ — 

A  Mixed  Expression  is  an  expression  which  has  both 
integral  and  fractional  parts;  as  a-\--,  or  ^  +  ^-3 — 

136.  We  have  by  §  30,  . 

ax(-  +  -)=ax-  +  ax-=b  +  c.         (§9,3) 
\a     aj  a  a 


4a2  +  15a6- 

-4?>2 

3aj3-17x2  +  4a;  +  4 

3aj3-14a)2- 

-llaj-2 

2a'  +  9a'- 

-2a-3 

6a3  +  23a2- 

-22a +  3 

m^  +  m^  + 

m  +  6 

m^  +  6m^4-6m  —  4 

a^-\-2a'x- 

■2aaf-x^ 

FRACTIONS.  101 

Kegarding  6  +  c  as  the  dividend,  a  as  the  divisor,  and 

-  4-  -  as  the  quotient  (§  54),  this  may  be  written 
a     a  ^ 

h  +  c_  6      c 

a     ~a      a 

137.  A  fraction  may  be  reduced  to  an  integral  or  mixed 
expression  by  the  operation  of  division,  if  the  degree  (§  108) 
of  the  numerator  is  equal  to,  or  greater  than,  that  of  the 
denominator. 

1.  Reduce  - — "*"    ^^~     to  a  mixed  expression. 

•^  ^        '  3x  3x       3x      3a;  3x 

12ar'  — 8a^-l-4a;  — 5 

2.  Reduce — to  a  mixed  expression. 

4ar^H-3  ^ 

4x2  +  3)12x«  -  8x2  +  4x  -  5(3a;  -  2 
12x8  +9x 

-8x2-5x 
-8x2  -6 

-5x+  1 

A  remainder  of  lower  degree  than  the  divisor  may  be  written  Over 
the  divisor  in  the  form  of  a  fraction,  and  the  result  added  to  the 
quotient. 

Thus,  12x3-8x2  4-4x-6^3^  _  ^  ^  -5x+l. 

4x2  +  3  4x2  +  3 

If  the  first  term  of  the  numerator  is  negative,  it  is  usual  to  change 
the  sign  of  each  term  of  the  numerator,  at  the  same  time  changing  the 
sign  before  the  fraction  (§  130). 

Thus,  12x«-8x2  +  4x-5  ^  3^  _  .,  _  6x^    ^^^ 

*  4x2  +  3  4x2  +  3 


EXAMPLES. 

Reduce  each  of  the  following  to  a  mixed  expression: 

3    12 a^- 16 a; +-7  ^    15  a» +■  6a^  -  3a  -  8 

4  a;  '  3  tt 


102 

ALGEBRA. 

5.   ^^  +  1                 6 

a^-\-f 

y    a3-263 

205  +  3                  ^• 

X- 

-y 

a +  6 

g     15a^  +  lla'-15a 

-6 

12 

12  m^  +  19  ^7^2  _  7  ^ 

3a4-4 

4m2  +  l 

9.        ^^^' 
2m  —  5/1 

13. 

ic^  +  2/'' 
05  +  2/ 

x^  —  x  —  1 

14. 

18a3-3a2  +  38 
3a2-4a  +  5 

^^    12a^-5a-5 

15. 

a^+6^ 

4a 


jg    8  a;^  4- 16  a^  -  10  a^  -  28  a;  +  11 

2aj2  +  a;-3 


138.  To  Reduce  a  Mixed  Expression  to  a  Fraction. 

The  process  being  the  reverse  of  that  of  §  137,  we  have 
the  following  rule : 

Multiply  the  integral  part  by  the  denominator. 

Add  the  numerator  to  the  product  when  the  sign  before  the 
fraction  is  +,  and  subtract  it  when  the  sign  is  —  ;  and  write 
the  result  over  the  denominator. 

1.    Reduce  — — h  x  —  2  to  a  fractional  form. 

2aj-3 

Wehave,     ^  +  5  ^  ^x +  5 +(x  -  2)(2a.  -  3) 

2x-3  2x-3 

2x-3 

2x-S 

If  the  numerator  is  a  polynomial,  it  is  convenient  to  en- 
close it  in  a  parenthesis  when  the  sign  before  the  fraction 
is  — . 


FRACTIONS.  103 

2.  Reduce  a  —  h to  a  fractional  form. 

a  +  b 

We  have,  ^  _  5  _  «^  -  «?>  -  ^^  ^  («  +  &)(«  -  ^)  -  («^  -  «&  -  f ) 
a  +  6  a  -\-  h 

a-\-b 
ab 


a-{-b 


Ans. 


EXAMPLES. 
Reduce  each  of  the  following  to  a  fractional  form : 

x  +  2y 

12.  4m'-9+^'"(^'"-^). 
2  m  4- 3 

13.  2a?  +  3a-i^^2^^. 
2a-l 

15.  .  +  ,  ^  +  ^ 


3 

a      1  1  "^"^"^ 

"      '"3a 

4. 

a;  — 2/ 

5. 

5a  1  1           ^      . 

2  a  -  3 

6. 

3^-2-11  ^^  +  7. 
5a; 

7. 

^      3a-6 

3a  +  & 

8. 

2             .     2        2n« 

m^  —  mn  4-n'' 

m  +  n 

9. 

2a  +  5a;      ^ 
2a-5a;       ' 

10. 

3.  +  4  +  ^f  +  15 

2  6* 


16.   a3_^a25_|_^^2^53^ 


17.  _(^zd}!__(ar^_a;H-l). 

18.  m4-3»i  — 


3  a; -4  m^-Smn-i-dn^ 

139.  To  Reduce  Fractions  to  their  Lowest  Common  De- 
nominator. 
To  reduce  fractions  to  their  Lowest  Common  Denominator 

(L.  C.  D.)  is  to  express  them  as  equivalent  fractions,  having 
for  their  common  denominator  the  lowest  common  multiple 
of  the  given  denominators. 


104  ALGEBRA. 

Let  it  be  required  to  reduce  ^^,  ^^,  and  ^^  to 

3  a^b    2  ab^  4  a^b 

their  lowest  common  denominator.    . 

The  L.  C.  M.  of  3a%  2ab%  and  Aa^b  is  12a^b'  (§  125). 
By  §  129,  if  both  terms  of  a  fraction  be  multiplied  by  the 
same  expression,  the  value  of  the  fraction  is  not  altered. 

4cd 
Sa'b 


Multiplying  both  terms  of  -^-j-  by  Aab,  both  terms  of 


-^  by  6  a^,  and  both  terms  of  — ^  by  3  b,  we  have 
2  ab^  4  a^b 

16  abed    18  a^mx        ^  Wbny 
12  a'b' '  "12^^'  ^^     12^2* 

It  will  be  seen  that  the  terms  of  each  fraction  are  multi- 
plied by  an  expression  which  is  obtained  by  dividing  the 
L.  C.  D.  by  its  own  denominator ;  whence  the  following  rule : 

Find  the  lowest  common  multiple  of  the  given  denominators. 

Divide  this  by  each  denominator  separately,  multiply  the 
correspondiiig  numerators  by  the  quotients,  and  write  the 
results  over  the  common  denominator. 

Before  applying  the  rule,  each  fraction  should  be  reduced 
to  its  lowest  terms. 

140.   1.   Reduce  -^^  and  - — — to  their  lowest 

a^  —  4  a^  —  5a  +  6 

common  denominator.  ^ 

Wehave,  a2_4  =  («_|.2)(a-2),  and  a'^-^a+Q  ={a-2){a-^). 

Then  the  L.  C.  D.  is  (a  +  2)(a  -  2) (a  -  3).  (§  125) 

Dividing  the  L.  C.  D.  by  (a  +  2)(a  -  2),  the  quotient  is  a  —  3  ; 
and  dividing  it  by  {a  —  2)  (a  -  3),  the  quotient  is  a  +  2. 

Multiplying  4  a  by  a  -  3,  the  product  is  4  a  (a  -  3)  ;  and  multiply, 
ing  3  a  by  a  4-  2,  the  product  is  3  a{a  +  2). 

Then  the  required  fractions  are 

^ "(«-«) and  3«(«±2) _,  Am. 

(a  +  2)(o-2)(o-3)  (a  +  2)(a-2)(a-3) 


FRACTIONS.  105 

EXAMPLES. 
Keduce  the  following  to  their  lowest  common  denominator : 

5x 


^     5xy    3xz     4:yz  g  3  a; 

~6"'   TT'     2r'  '    6ar^-f-2ic    9a^-l 

g        1  2  6  y      aa;  6?/^  ca;^!/ 

2m^w     3??i7i^    ^m^n^  '    x-\-y     (x-\-y)^    (aj+y)* 


4. 


2a-5c^    4a  +  36  q        2a  4:b^ 


da'b  12  ac^  a'-b'    a^-\-b^ 


K     Ta^^      9  6y      _8c^  g        3  6 

10 


82^2-    iOa^2     i52/;22  a  +  1     a-1     a^  +  1 

X2 


11. 

12. 


3a;«-12ar^    x'-Qx-\-d>    3?-% 

x-{-y a  —  b 

ax  —  bx  —  ay  -\- by    a^  —  2xy  -^  y^ 

a  +  5  a +  3  a  — 2 


a^-a-6    a^  +  Ta-hlO     a^-\-2a-15 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 


141.  We  have  by  §  136, 


In  like  manner, 

a     a 


6  ,  c  _  b  -\-c 
a     a        a 

b     c      b  —  c 


Whence  the  following  rule : 

To  add  or  subtract  fractions,  reduce  them,  if  necessary,  to 
equivalent  fractions  having  the  lowest  common  denominator. 

Add  or  subtract  the  numerator  of  each  resulting  fraction, 
according  as  the  sign  before  the  fraction  is  -\-  or  —,  and  write 
the  result  over  the  loivest  common  denominator. 

The  final  result  should  be  reduced  to  its  lowest  terms. 


106  ALGEBRA.  ^XL 


142.1.  Simplify  i|^+i^.        /;i;<^^ 

The  L.  C.  D.  is  12  a^b^ 

Multiplying  the  terms  of  the  first  fraction  by  3  b^,  and  the  terms 
of  the  second  by  2  a,  we  have 

4  a  +  3     1  -  6  ?>-^  ^  12  ab^  +  9  b^  _^2  a- 12  ab^ 


^a%  6ab^  12  a'b^  12  a'^b^ 

^  12  ab-^  +  9  b^  +  2  g  -  12  gft^  ^  9  ft'^  +  2  g 
12a2fe3  12  a^b-^  ' 


J.WS. 


If  a  fraction  whose  numerator  is  a  polynomial  is  preceded 
by  a  —  sign,  it  is  convenient  to  enclose  the  numerator  in  a 
parenthesis  preceded  by  a  —  sign,  as  shown  in  Ex.  2. 

If  this  is  not  done,  care  must  be  taken  to  change  the  sign 
of  each  term  of  the  numerator  before  combining  it  with  the 
other  numerators. 

2.    Simplify  g-^ j^. 

The  L.  C.  D.  is  42. 

5x-4?/     lx-2ij     35X-28?/     21x-6.v 


Whence, 


6  14  42  42 

_  35  a;  -  28  y  -  (21  x  -  6  y) 
-  42 

35«-28?/-21x  +  6?/ 


42 

Ux-22y  _  7x-  11  y 
42         ~        21 


,  Ans. 


EXAMPLES. 

Simplify  the  following : 

3     5ct-6     3a  +  7  g     3x-\-4:     2x-\-5 

8  12     *  '        12  16 

-46  c    a  —  4:X     7x  —  6a 


Sxy^     5a^y  6ax^  Oa^a; 


FRACTIONS.  107 

„     x  —  3m  .  4:X-\-m  g     2a— 9    8a— 5    4a+T 

'      24  m  32  a;    *  "        7  14  28~' 

«     2a  — b     2b  — c    2c  — a       .^    x-^1     3a;— 4     5:^+7 
ab  be  ca  2x         bx^         8a;^ 

jj     5a  +  l     26  +  3     7c-4 
Qa  86  12c 

12  3g;-y     4a;-5y     6a;^  +  2y^ 

5a;     "•"     10?/  15a;3/     ' 

13  6a;  +  l      5a;-2     8a;-3      7a;  +  4 

3  6  9  12     * 

14  3a  +  4     4a-3     5a  +  2     6a-l 


•  K    2ft  —  36     3a  +  6     4ft  —  56      5  a -f76 
9  18  27  >  36      ' 

1 


16.    Simplify 


01?  +  X        'J? 


We  have,  cc-  +  x  =  x(x  +  1),  and  x^  —  x  =  x{x  —  1). 
Then  the  L.  C.  D.  is  x(x  +  l)(a;  -  1),  or  x(x2  -  1). 
Multiplying  the  terms  of  the  first  fraction  by  x  —  1,  and  the  terms 
of  the  second  by  x  -f  1 ,  we  have 

1 1      ^    x-l  x  +  1 

X2  +  X       X2  -  X       X(X2  -  1)        X(X2  -  1) 

^x-l-(x  +  l)^x-l-x-l^      -2         ^^^^ 
x(x2-l)  x(x2-l)         a;(x2-l)' 

By  changing  the  sign  of  the  numerator,  at  the  same  time  chang- 
ing the  sign  before  the  fraction  (§  130),  we  may  write  the  answer 
2 
x(x2-l)' 

Or,  by  changing  the  sign  of  the  numerator,  and  of  the  factor  x2  —  1 

2 
of  the  denominator  (§  131),  we  may  write  it 


x(l  -  x2) 


108  ALGEBRA. 

17.    Simplify  -^4:-^  -  ...       f  „   ,   o  +  TF      ^ 


Wehave,  a2_3a+2  =  (a-l)(a-2),  a2-4a  +  3  =  (a-l)(a-3),  and 

a2-5a  +  6=:(a-2)(a-3). 

Then  the  L.  C.  D.  is  (a-  l)(a  -  2)(a  -  3). 
Whence,  — ^ — + 


a^-Sa-\-2     a2_4rt^3     ^^-Sa  +  G 

0-3 2(a-2)  a-1 

(a-l)(a-2)(a-3)      (a-l)(a-2)(a-3)      (a-l)(a-2)(a-3) 

a-8-2(a-2)+a-l_a-3-2a  +  4  +  a-l_Q    ^^^-^ 
(a-l)(a-2)(a-3)         (a  -  l)(a  -  2)(a  -  3) 


Simplify  the  following : 

18.         ^        I        ^      -  23.    ^  +  ^^m^^. 
3  a  +  5      4  a  —  7  in  —  n      m  -\-  n 

jg    __m 1^^  24     1  — ^    _^  +  ^, 

m  — 1      m  +  l  l+ic     1— ic 

20    -^^ i-.  26.   4a^  +  l_2a-l. 

■    2aj  +  l      5x-6  4a2-l      2a  +  l 

21,     ^    I     ^ 26.  ^^-2/    y(y-^^). 

'    a-{-b      a  —  b  x  x^  —  xy 

22      ^^    _ 2a^  — 6a  — 3  27        ^  +  ^  ^  ~"  ^ 


aH-4 

a2_3a-28 

4:0" -9b'     (2 a  4- 3 6)2 

28. 

1 

a^  _l_  4  a;  -  12 

1 
c^-Sx-54: 

29. 

x'-6ax-\-9a^ 

X 

!     a;2  +  4  aa?  -  21  a2 

30. 

a'-\-b' 

a         b 

32        a            6            2  62    ^ 

a^-^ab 

a  +  b     a 

a  —  6      a  +  6      a'  —  b' 

31. 

^     + 

3x         6a^ 

33.    *      -^     1. 

1  +  a;     1  —  x     1— a^  a;  —  2/     a;  +  3/ 


38. 


34. 


FRACTIONS.  109 

1         .      2x 


a{a  +  x)      a(a  —  x)      or  —  x^ 


35        1  2x         3a^  +  4 

'    x  +  2      {x^2f     {x^2Y 

36.    ^ 1 i.  39.    -i (^^-^n 

a-3     a  +  6     a  2a4-&      Sa^^-^' 

qy     a;  4-2  _  a?  —  2  _     16  ac\^-^^     a  —  x        4:  ax 


x  —  2     x-\-2     a^  —  4  a  —  X     a-\-  x     a-  —  x^ 

x-\-y     x'  +  f  ^^     ^      2(x-\-y)  ^  (x -j- yf 

X  —  y     x^  —  y^  x  —  y        (^  —  2/)^ 


43. 


m  —  ?i      (m  —  ?i)^      (m  —  Jii)-' 

x-\-\      a;  —  3  a;  —  5 

x  +  2      a;-4     a^-2a;-8" 


44  1  I  1  I  1 

(a  -  6)  (6  _  c)      (6  -  c)  (c  -  a)  ^  (c  -  a)  (a  -  6) 

45. + 


a;-3     a;-2     a^-5a;  +  6 
46.-1^+     1  2a 


47. 


a  +  6  '  (1-6      0,2  +  62 

1 3a;  oa; 

a  —  a;     a^  —  m?     (f  —  x 


4g    ^^ g  a^-A 

*    a  4-1      a^-a  +  l      a^  +  l' 


49. 


a;  +  z ?/  +  g .T  +  ?/ 

(^-y)(y-^)    (p^-y)(x-^)    (^-^)(y-z) 


50         a;  +  2  2(a;-l)  a;-3 

a^  +  4a;  +  3     a^^x-6     af-x-2 

In  certain  examples,  the  principles  of  §§  130  and  131 
enable  us  to  change  the  form  of  a  fraction  so  that  the  given 
denominators  shall  be  arranged  in  the  same  order  of  powers. 


110  ALGEBRA. 

51.    Simplify  _A_  +  25ia^ 
^     ^   a-b      h^-oj" 

Changing  the  signs  of  the  terms  in  the  second  denominator,  at  the 
same  time  changing  the  sign  before  the  fraction  (§  130),  we  have 

3         2h  +  a 
a-b     cfi-  62* 
The  L.  C.  D.  is  now  d^  -  b^. 

Whence,     -^ 2b±a^S{a  +  b)-(2h  +  a) 

a-b     a'^-b'^  a^  -  h'' 

_  3a  +  3&  -26  -  g  _  2  a  +  6      . 


52.   Simplify 


111 


{x-y){x-z)      (y-x){y-z)      (z-x)(z-y) 

By  §  131,  we  change  the  sign  of  the  factor  y  —  x  in  the  second  de- 
nominator, at  the  same  time  changing  the  sign  before  the  fraction ; 
and  we  change  the  signs  of  both  factors  of  the  third  denominator. 

The  expression  then  becomes 

1,1  1 


(x-y)(x-z)      (x-y){y-z)      (x-z)iy-z) 
The  L.  C.  D.  is  now  (x  —  y){x  —  z) (y  —  z) ;  whence  the  result 

-(y-^)-\-(^-^)-(^-y)  ^  y-z+x-z-x+y 

{x-y){x-z){y~z)        (x-y)(x-z)(y-z) 

2y-2^  ^  2(y-z)  ^  2  ^^^ 

{z-y){x-z){y-z)      (x-y){x-z)(y-z)      (x-y){x-z)' 

Simplify  the  following: 

53.  -y — ^-^.        57. 

Qir  —  xy     y^  —  xy 

54.     ^-  +  5  _2x-l_  5g 

3a;-6     8-4a! 

a^  —  9     3  —  a 
56.   _J--  +  _1_.  60. 


a 

1   1 

1 

ab-W 

b  —  a 

b 

a 

a 

2 

a  +  l  ' 

1-a 

a?-l 

X 

X 

^   , 

2  +  x 

2-x 

a^-4 

X 

y 

2/ 

4  w  —  m^     m^  — 16  ^  +  2/     a;  —  1/     y'^  —  o^ 


FRACTIONS.  Ill 

61     1  _      1        1    ^g'^  — 9       62        ^  ^  ^  ^^ 


63. 


a     2a -S     9a- Aa^  m  +  2     m-2     i-m^ 

1  1 


(a  -  6)  (a  +  c)      (&  -  a)  (6  +  c) 


64.    -^+       2a.  1 


65.   , 1 , t ,+ 


(^-y)(y-^)    {y-x)(x-z)    {z-x)(x-y) 

66. ^^; + 


(a-b)(a-c)      (b-c){b-a)      {c-a)(c-b) 


MULTIPLICATION  OF  FRACTIONS. 

a  c 

143.  Required  the  product  of  -  and  -• 

0  a 

Let  ?x^  =  a;.  (1) 

b     a 

Multiplying  each  of  these  equals  by  6  x  c?  (§  9,  1),  we  have 

-  X-  X  b  X  d  =  X  X  b  X  d. 
b     d 

Or,  since  the  factors  of  a  product  may  be  written  in  any 
order, 


f^  xb\xf^x  d\=x  X  b  X  d. 


Whence,  (a)  x  (c)  =  x  x  b  x  d.  (§  9,  3) 

Dividing  each  of  these  equals  by  6  x  ci  (§  9,  1),  we  have 
axe 


b  X  d 


(2) 


From  (1)  and  (2),  f  x  |  =  ^.  (§  9,  4) 

^  b     d     b  X  d 

We  then  have  the  following  rule  for  the  multiplication  of 
fractions : 

Multiply  the  numerators  together  for  the  numerator  of  the 
product,  and  the  denominators  for  its  denominator. 


112  ALGEBRA. 

Common  factors  in  the  numerators  and  denominators 
should  be  cancelled  before  performing  the  multiplication. 

Integral  or  mixed  expressions  should  be  expressed  in  a 
fractional  form  (§  §  135,  138),  before  applying  the  rule. 

144.   1.  Multiply^  V  1^.  -^. 

We  have    «^^^y<r^W>^  2  x  5  /H  a^hH^_  UH     . 


^bx"      ^d^^     32  X  22  X  a8&x2y8 
In  this  case,-Hhe  factors  cancelled  are  2,  3,  a^,  b,  x^,  and  y. 

2.   Find  the  product  of  — -,  2 ^^,  and  x^  —  9. 

a^  +  fl7  — 6  x  —  3 

We  have,  ^ x  f2 -^^:i^^  x  (x2  -  9) 


x^  +  x-6  x-S 


X 


_  X  ^  X  (X  +  3)  (X  -  3)  =  ^^^±11,  Ans. 
(x  +  3)(x-2)      x-3     ^         ^^  x-2   ' 

In  this  case,  the  factors  cancelled  are  x  +  3  and  x  —  3.  . 

EXAMPLES. 

Simplify  the  following : 

3     l^^^Txv'  8     27  m^        15n^         7  x' 

UxY        ^'  20n'x     2SxY     ISm'y 

,      6a^m      20  6V  o     a^-\-a-SO  ^^        5  a 

^'     — :  X  — — :: — r*  *'• 


25  5V      3aV  3  a  a^-4.a-5 

K     5x      3y_     Sz  ^Q      9m^  — 1       m^-\-5m 
^y     lOz^x  '    m3-25m'    3m-l' 

«     4a^     156^      21c^  ,,     a^-h3a;-18      2a.-8-4a^ 
96^^  7c^       lOa^*  '    aj2_8x4-12       a^^  -  36  * 

y.  3a^&%  6  5V      lOc^g  -o     xy +  y^     x"  ^- xy -2y^ 

4c*        5a«        96^  ^'*-    0?  -  xy^  x" +  2xy +  y^ 


13. 


FRACTIONS.  113 

a^-ab~6b''  ^  a'+^ab         '    ar^H-8       a^-^a^-\-x' 
-.     5.^•-f-15       3a;-9  ^  8ar-2 

AO.      -— —  X   T7. z  X 


8a; -4       10a; +  5     3ar^-27 

Ifi        a^  -{-2  a         a- —  16  a^  -\-  a 

a^  —  3  a  —  4      a?  —  a      a^  +  6  a  +  8 


17.    f6     2-^  +  3 


■'   A       2^  +  33// 


•    (x-yf-z'     a?-{y  +  zf 
19.    ±^Ax2-:^'x^  2«6 


a' +  6*      «  +  <'      V       a''  +  a6  + 


^> 


20         g^a;  +  ox'^  g^  -h  2  oa;  +  a;^     g''  —  2  ga;  +  ag^ 

g*  —  2  g  V  +  a;*  g^  -h  ar*  ga; 

'*^*   16a;*-9ar'''2ar^  +  2'^V  ^a;-iy 

DIVISION  OF  FRACTIONS. 

145.  Required  the  quotient  of  -  divided  by  -• 

b  d 

Let  1^1  =  ..  (1) 

Then  since  the  dividend  is  the  product  of  the  divisor  and 
quotient  (§  54),  we  have 

b     d 
Multiplying  each  of  these  equals  by  -  (§  9, 1),  we  have 

2x^  =  ^xxx^  =  «.  (2) 

bed  c 

From  (1)"  and  (2),      ^  ^  ^  =  ^  x  -•  (§  9,  4) 

0     d     0      c 


114  ALGEBRA. 

Therefore,  to  divide  one  fraction  by  another,  multiply  the 
dividend  by  the  divisor  inverted. 

Integral  or  mixed  expressions  should  be  expressed  in  a 
fractional  form  (§§  135,  138)  before  applying  the  rule. 

146.   1.   Divide  — —  by  q-r^n- 
5  oi?y^         10  x-y' 

5xV     i0a;2i/7     5a;V      9«^'''      Sft'^x' 

2.   Divide  9+^  by  3  +  ^. 
We  have, 


«-^)-(^-.-^) 


_9x2-4y2        a; -2/    _  (3ic  + 2y)(3x -2y) 
aj2-y2    ^3x  +  2^~      (a;  +  y)(x-«/) 

3x-2w      , 

= -,  Arts. 

x  +  y 

EXAMPLES. 

Simplify  the  following : 

3.   ?i^^8a362.  7.    ^^_?V^|^  + 


-     21  an^  .  14 aV  g    a^  +  lQa  +  21  .   a^-9 

5  3.2  g    a?^+4fl?y+4y^  .  xy-\-2y\ 

x^—ex-\-S  '  Q(?—x—V2,  '  x  —  y         '   Q^  —  xy' 

g    4m^-25n^  .   2mn-hn^    ^^    y?  -  x  .  a^-2a;  +  l 
j^  a3_8         ^  a2-^2a  +  4 


12. 


a2  +  7  a  +  10         a^  +  2  a 
a2_5^5_1452  ^  a2_3a5-286^ 
a2  4.5a6-2462  '  a' -  S  ab -{- 15  b'' 


13. 
14. 


FRACTIONS. 

(.. 

a- 
a- 

-5x 
-2x^ 

a2- 

-b'- 

-(^-2  be  .  a 

-b 

—  c 

115 


a2-62_c2-4-2dc     a  +  6.^c 


COMPLEX  FRACTIONS. 

147.  A  Complex  Fraction  is  a  fraction  having  one  or 
more  fractions  in  either  or  both  of  its  terms. 

It  is  simply  a  case  in  division  of  fractions,  its  numerator 
being  the  dividend,  and  its  denominator  the  divisor. 

14a  1.   Simplify  — ^.  ,    ~r     JC 

We  have,      -!L- = -J!— =  a  x  :rj—  (§146)=-^,  AiisT^^ 
,       c     bd-c  bd  —  c^"        '      bd  —  € 


It  is  often  advantageous  to  simplify  a  complex  fraction 
by  multiplying  its  numerator  and  denominator  by  the  lowest 
common  multiple  of  their  denominators  (§  129). 


2.   Simplify  —J- —' 


a  —  b     a-{-b 
The  L.  C.  M.  of  rt  +  6  and  a-b  is  (a  +  6)  (a  -  6). 
Multiplying  both  terms  by  (a  +  6)(a  —  6),  we  have 

a(a  +  b)-  a(a  —  b)  _  a^  -{-  ab  —  a^ -\- ab  _    2  ah      ^ 

b(a  +  b)-ha(a-b)      ab  +  b^  +  a^ -  ab~ a^  +  b^' 

EXAMPLES. 
Simplify  the  following : 

a      c  1,1  1 

r  — 3  ^~^o —  ^  —  ::^ 

^    b      a  .  2m  _  ar 

o.    •  4.    T — •  0.    T— 

b      d  4m  X 


116  ALGEBRA. 

™_:!L  ?+l_^  ^-13+^ 

mn  y  X  x 

^_2-i-^  a_a^-6  a;         a?  —  ^ 

,y    % 2^^  ^^    5      a  +  6  jg    a;4-y         a^ 


23  b  ,  a  —  b  x         x-\- 


y     X                     a     a  +  b  ^  —  y         ^ 

a-x                     ^     82/  26^ 

a-- ~^  +  ^-  ^ Z 

8 i±^.  11.  y    \  ■  14.  — ^. 


a^  —  ax 


\^- __2  +  ^i  a  + 


l-faa?  2/  ^  a  +  25 

a^  +  a  +  1  H — r  -I- 


15.   T-^— ^-  16. 


a  +  36     a  — 36 


1  2a4-6      2a-6 

^■^a-1  a-36     a  +  36 


17.    Simplify    r 

1+  — 


-*\ 


Wehave,  __^  = —^  = -^.^^  =  ^^^^ ,  ^«.. 
1  a,+  l 

X 

In  examples  like  the  above,  begin  by  simplifying  the  lowest  complex 

1  X 

fraction;  first  multiply  both  terms  of by  x,  giving  ., ,  and 

1  +  1  ''+' 

1  "^  X  4-  1 

then  multiply  both  terms  of  bv  x  +  1,  giving ^- . 

1   I      ^       '  X  +  1  +  x 

X  +  1 
Simplify  the  following : 

2  1 

18.     3 i-j—  19.     1 -rr—' 

5+     ^  -  ' 


7+-  3- 


X 


FRACTIONS.  117 

^      S{a'-\-b^  x^y     x^  +  f 

20.  ^^'-^'  23       ""-y     "^-^ 


^      2  (g  +  2  6)  ■       x"        xjx'  +  y^) 

3a  +  6  x  +  y       {x  +  yf 

x-\-  a      X  —  a  2n(m  —  n) 

x  —  a     x-^a  g.                         m  +n 


or*  +  a^  _  -l  '  ?^^  H-  ?i- 


l-x"  l-i-x"  g  +  6  _  g^  +  &« 

1  +  ar^  1-a^  g  -  6     g«  -  6« 

1-a;  1  +  a;'  a  +  6     g«  +  6«' 

l  +  a;  1-ic  a-6"'"g3-63 

MISCELLANEOUS  AND  REVIEW  EXAMPLES. 
149.  Reduce  each  of  the  following  to  a  fractional  form : 

4g  +  3  d^-ab  +  b^      ^  ^ 

Simplify  the  following : 

Q  1  2g  ,        Qdx 


2a-Sx     (2a-Sxy     (2a-3xy 

4         (l-x)(l-y)  5    (a^-2Y-a^ 

(l-^xyy-(x-\-yy  '    a*-3g'^-4 

ft    oi-\-b  .  c-{-d        2 (gc  —  bd) 

a  —  b      c  —  d  (b  —a){c  —  d) 

g    6xy-(x  +  2yy  .q    (a^  -  6 a;  -  4)^  -  144 
a?  +  Sf       '  '     (a^  +  a;-ll)2-81  ■ 


118 


11. 


ALGEBRA. 

3a 


4a 


3a  +  2a2-7a4-10     a^-Ga-f-S 


13. 
14. 


12.   (2ai-S-,^^(2a-3  +  -^^. 
\  2a-\-3j\  2a -3j 

oc^  -{-y^     Qi?  —  if  yf-     (^_^\y 

x^  —  y^     a?^  4-  2/^ 


^  +  ^ 


15.      :^_-  +  ^_ 

a^  H-  ci'^h  —  a^h^  —  ah^        jg    ahd  +  abd^  —  ah(?  —  6^cc? 
a'h  -  a^b'  +  a6^  -  b''  '   ahd  -  abd^  -  abc^  +  b'cd 


17.   f^  +  -^ 
\fl;  —  2     a;  — 


18. 


19. 


20. 


Sj\Sx-S     x-\-2 
1    \     /  1 


x^-^3x-\-2     x+1 


3a;  +  2 


x-\-2j 


m 


m 


2{m  —  n)     2(m  H-  n)     iii?{w?  —  inF) 

(g  4-  ^  +  c)^  -  (a  +  6  -  c)^ 
(a  -  6  +  c)2  -  (a  -  6  -  c)2* 

2a(2a-3&a;)+ 36(3  &  +  2aa;) 
(2a-36x)2+(36  +  2aa;)2 

a?  — y  ,  a;  +  g  _      (y  +  g)^ 

aj+g      ic  —  2/      (ic  —  2/)(a;  +  z) 
a'b  -  ab^ 


b- 


23. 


(a-\-by 


a-{-b  ,  a^  +  d^ 


24. 


a—  b 
1 


25. 


aV  a?  —  ct     a;  + 


— V 

-2ay 


4a:^-16a^  +  17a;-3 
6a3-17a;2  +  8a;  +  6* 

3 


a?^  -h  aa;  —  2  a^ 


6a6  +  962  _  /g2_95: 


a6  —  6  6 


a2_4^5_j_452     \^a2-462     a'-^ab-Qb 

gy    a;^-2a^y^  +  /  Y^' ~ '^  ♦  (a;  +  y)^-2a^y 
a;*  +  2a^/  +  2/*  '  V    ajy      *  a;y 


5' 


FRACTIONS.  119 

nn    m^  -f  2  7nn  +  4  n^     4  ?>i^  —  9  n^  _^  2  m  -\-Sn  . 
2  m  —  3  n  m^  —  Sn^        m'-  —  4  n^ 

x-[-y     a^  —  f 
1  I  1  31.  "'"^     "^"^^ 


a(a  —  6) (a  —  c)      6(6  — c) (6  — a)         '  a.-^  +  2/^  _  ic  —  2/ 

a;3  _  2^     a;  +  2/ 

(^-y)(^-^)      (y-2;)(2-a;)      (2-«)(a;-y) 

33    A  ^^         Vi ^^         ^  >c  ^'^  "^  ^^ 

•    ^^       a2-a6  +  6VV       a'  +  2a6  +  6V     «' -  6^ 

34.       14-^-  +  .-^+      ^ 


14-251+3^1  + a;' 

(First  add  the  first  two  fractions ;  to  the  result  add  the  third  frac- 
tion, and  to  this  result  add  the  fourth  fraction.) 

35.    -H -^r+   „      .  + 


a-2a  +  2a2H-4a*  +  16 
36.       1  1.1  1 


-1     x-^1      x  —  2     a;4-2 

(First  combine  the  first  two  fractions,  then  the  last  two,  and  then 
add  these  results. ) 

37        1  1  2a  2a 


a  -  6     a  -f  6     a^  -  6^     a^  +  6^ 
38        1  1  3ar^  3a^ 


x-1     x  +  1     aj3  +  l     a^_l 
39  4a;  — 3  2  a;  —  5 


6ar^  +  13  a;  -  5     12 ar^  H- 5a;  -  3 
(Find  the  H.  C.  F.  of  the  denominators  by  the  method  of  §  117.) 
^Q         3a  4- 2  5a -1 


6a2- a- 12     10a2-19a  +  6 


120  ALGEBRA. 

XIII.  SIMPLE  EQUATIONS  (Continued). 

SOLUTION  OF  EQUATIONS  CONTAINING  FRACTIONS. 

150.  Clearing  of  Fractions. 

^.j,i  ..      2x     5     5x     9 

Consider  the  equation = -• 

^  3       4       6       8 

The  lowest  common  multiple  of  3,  4,  6,  and  8  is  24. 

Multiplying  each  term  of  the  equation  by  24  (§  71,  2), 

we  have  ^^      ^^        ^ 

16aj-30  =  20a;-27, 

where  the  denominators  have  been  removed. 

We  derive  from  the  above  the  following  rule  for  clearing 
an  equation  of  fractions  : 

Multiply  each  term  by  the  lowest  common  multiple  of  the 
given  denominators. 

151.  1.    Solvetheequation  ^-1  =  ^-^. 

6      3,     5      4 

The  L.  C.  M.  of  6,  3,  5,  and  4  is  60. 

Multiplying  each  term  of  the  equation  by  60,  we  have 

70a;-  100  =  36a; -15. 
Transposing,  70  a;  -  36  a;  =  100  -  15. 

Uniting  terms,  34  a;  =  85. 

Dividing  by  34,  ^  =  It  =  %  ^^*- 

EXAMPLES. 
Solve  the  following  equations : 
2.   »+|_^  =  9.  3.  ^-^  +  ^^0. 


9. 

3a;  5  x  7x  4x 
2       14     3  ~  6        7  ' 

10. 

2x  X  3  _7x  Sx 
5       2     10       8        4 

11. 

2  3  4  5  _  1 
3a;     4a;     5u;     6a;     20 

^^JPALirofy#^  SIMPLE   EQUATIONS.  121 

4     ^_?  =  ^  +  l  8     -5 L  =  1_A 

'234       8*  '    18a;     6x     4     9a;* 

1^      5     i£_?  =  ^_^ 
'9       3       6        2  * 

6     '^'^     4a;2a;_      11 
■     2        3        5  ~      6* 

^      7.    A_i  =  l__i.. 
5a;  10     4a; 

If  a  fraction  whose  numerator  is  a  polynomial  is  preceded 
by  a  —  sign,  it  is  convenient,  on  clearing  of  fractions,  to 
enclose  the  numerator  in  a  parenthesis,  as  shown  in  Ex.  12. 

If  this  is  not  done,  care  must  be  taken  to  change  the 
sign  of  each  term  of  the  numerator  when  the  denominator  is 
removed. 

12.  Solve  the  equation  ?^fll  -  i^  =  4  +  ^4±i. 

4  5  10 

The  L.  C.  M.  of  4,  5,  and  10  is  20  ;  multiply ing  •  each  term  by  20, 
we  have 

16a;-5-(16x-20)  =  80+  14x  +  10. 

Whence,         16x  -  6  -  16x  +  20  =  80  +  14x+ 10. 

Transposing,       15  x  -  16  x  -  14  x  =  80  4  10  +  5  -  20. 

Uniting  terms,  —  15x  =  75. 

Dividing  by  —  15,  x  =—  5,  Ans. 

Solve  the  following  equations : 

13.  4x  +  »''-^^  =  ^.  16.    a,-3^±I  =  8^Ili-l. 


14.    S^     2.-2^^     2. 
3           9 

17. 

2x-5     3a;-8         2 
7               6              3 

15.    2.-^-^^^-^-  +  l. 

18. 

a;-|-2_  9       3a;  +  14 
10        35          14 

.      19    nx  +  i     14 

a;-f-3 

10a;4-7      ^ 

^                   2 

4 

8 

122  ALGEBRA. 


2  8^^  12 

5^  ^         20  5  2 

22     1^(^  +  ^)      5a; -4     5a;  +  12^^T 
9  12  6       ""    '' 

23.  li^-l(8^-5)  +  l(10a;-7)  =  ^(i^L±ll. 

^  O  D  4 

24.  ll^L^-l(3a)-l)  =  lIi-+I-?(7a;-2). 

3  2^  ^6  9^  ^ 


25. 


2a;  +  4      7a;- 1  ^13.-c  +  5      lla;-3 
5  2  3  10      ' 


oc     7a;  — 8      7a;  +  6      x-5     4a;  +  9 

«o. ■ ■ -  = *-• 

14  4a;  2  7a; 

o-     3(a;-3)      2(a;^-5)      5a.'^- 12^     9 
2  3a;  Gx"  2 

2  5 

28.    Solve  the  equation 


x-2     x-^2     a;2-4 

The  L.  C.  M.  of  X  -  2,  x  +  2,  and  x-^  -  4  is  x^  -  4. 
Multiplying  each  term  by  x^  —  4,  we  have 
2  (X  +  2)  -  5  (X  -  2)  -  2  =  0. 
Whence,  2x  +  4  -  5x  +  10  -  2  =  0. 

Transposing,  2x  —  5x  =  —  4  —  10  +  2. 

Uniting  terms,  —  3x  =  —  12. 

Dividing  by  —  3,  x  =  4,  Ans. 

If  the  denominators  are  partly  monomial  and  partly  poly- 
nomial, it  is  often  advantageous  to  clear  of  fractions  at  first 
partially;  multiplying  each  term  of  the  equation  by  the 
L.  C.  M.  of  the  monomial  denominators. 


29.    Solve  the  equation 


SIMPLE   EQUATIONS.  123 

6x  +  l       2.T-4      2a;-l 


15         7x- 16 


The  L.  C.  M.  of  the  monomial  denominators  is  15. 
Multiplying  each  term  by  16,  we  have 

,  30x-60^,^_3 

7  X  -  16 

Transposing  and  uniting  terms,     4  =  — ^-^ 

7x  —  16 

Multiplying  by  7 x  -  16,    28 x  -  64  =  30x  -  60. 

Transposing,  28  x  -  30  x  =  64  -  60. 

Uniting  terms,  —  2  x  =  4, 

Dividing  by —2,  x  =  —  2,  ^ns. 

Solve  the  following  equations : 

30.    -^ ^  =  0.         35.    I^±li^-24__A_=7. 

6x-\-2     3x-\-4:  (x-\-iy         ic+l 


3«-4     6a;-l  6  2(2a;+l) 

3a^  +  6a;4-4      '^'  *    3      3a;-4  9 

33       6a;  +  5    ^3a;-2  38         "  o  i 


2x(x  —  1)      ar^  —  1  3  if  —  5     x  —  2     x  —  S 

34       3a;  2x    ^2x-—5    gg    2a;-fT     5a;  — 4^a;  +  6 

2a;-h3     2a;-3     4ar^-9*  14         3a?+l         7    * 


40. 


3  4 


2a;-l     3a;  +  2 


41  2(.T-7)         a;-2     x-\-3^q 
x'  +  Sx-2S     x-4:     a;  +  7 

42  ^^  +  ^  _  4a;  +  7  _ 3a;  — 2 _ ^ 


43. 


6  6a;  +  ll  3 

1.3  6 


2a;  +  3     3a;-2     4a;  +  l 


124 


44. 
45. 

2x 

+  1 

ALGEBRA. 
2a^-l           9 

a.- +  17 

2x 
X  — 

-16     2ic  +  12      x^- 
•2      x-1      2x-\-4._ 

-2a; -48 

:0. 

a;  +  2     x-\-l      x^  —  1 

^g     2  +  3a;     2-3a;^36-4a; 
3-x        3  +  x  ~   x'-9  ' 

--     2a^  +  3a;-l      2a^-3a;  +  l_^ 

(^+l)(a?  +  3)_a;-6  7(3a^-8)      3(a;-2)_ 

(a.4-5)(aj  +  7)      x-{-2'  3(a!-3)  "^  3aj- 1 

49     3a^+5a;-4^3a;+5       ^^     2x±7__3^--5^r[x±2 
4.x'-3x  +  2~4.x-3  '    6x-4:     9x+6     9a^-4* 

2     a?- 3     oj- 5     X  —  6 


48. 


52. 


3  X-4: 


(First  combine  the  fractions  in  tlie  first  member ;  tlien  the  fractions 
in  the  second  member. ) 

eg    Sx-{-5     3x-2^6x-5        Tx-^S 

7  14  28  4(4  a; -3)* 

SOLUTION  OF  LITERAL  EQUATIONS. 

152.  A  Literal  Equation  is  one  in  which  some  or  all  of 
the  known  quantities  are  represented  by  letters ;  as, 

2x-\-a  =  bx^-10. 

153.  1.  Solve  the  equation  (6  —  cxy—  (a  —  cxy=  b(b  —  d). 
Performing  the  operations  indicated,  we  have 

62-2  bcx  +  c2x2  -  (a2  _  2  acx  +  c^^)  =  62  _  ab. 
Whence,  b'^-2  bcx + 02^2  -a^+2acx-  d^^  =  b^-ab. 
Transposing  and  uniting  terms,  2  acx — 2  bcx  =  a^  —  ab. 
Factoring  both  members,  2  cx(a  —  b)=  a(a  ^  b). 

Dividing  by  2  c(a  -b),  x  =  ^f^^  =  §-^,  ^»»- 


SIMPLE  EQUATIONS.  125 

EXAMPLES. 
Solve  the  following  equations  : 

2.  a(3bx-2a)=b(2a-3bx). 

3.  (x  +  ay  +  (b  +  cy  =  (x  -  ay+Q)  -  cy. 

-     X  —  a  ,     2x       r,  a    ^  —  ^     b  —x     2x     b 

4. 1 =  o.  o. = • 

X         X—  a  ax  a  a      X 

^     Sx  —  4      5  7n  —  2n  „  -,      x  —  2       1 

0.     = •  7.     X  rr  i- =  — 5* 

3x-\-4:     5m-\-2n  m         mr 

g     5x-2a     9x-5a^     S(x-^2a^)     5x^^ 
2a  3a«  a^  6a 

q     ax  —  b      bx-{-a _2  ,  a  —  b 
bx  ax  abx 

10.  2(x-b){2a-3b-3x)-(2a-3x){b  +  2x)=0. 

11 .  (x-\-  m) (x  -\-  n)  —  (x  —  m) (x  —  n)=2{m-\-  ny. 

12      x~^        x-{-b  _4:a^  —  b^ 
x  —  2a     X  -\-2a     ar*— 4a^ 

13     2x-\-3a^3x-\-A:b  -^     2nx  —  3^^     9nx  +  2 

2x—3b     3a;  — 4a  na;  — 1  37ix  —  l 

1-     a  —  b.b  —  c.c  —  a      ^ 
X  —  c     x  —  a         X 

16.    (x-i-ay-\-(x-ay  =  2x(x'-a^)-24:a\ 

,-     3  x(a  —  b)      a  —  2b     a  —  b  _r. 
s?  —  h^          x  +  b       b  —  X 

18     a?     g  —  2  5ca;  __  5  a;     8  ac  —  8  ^o;  —  0  a 
'    2  ~      4  6c      ~  6  c  12  6c 

19.    (a;  -  2  a  -f  3  6)2  -  (a;  -  2  a)  (x  +  3  6)  -  6  a6  =  0. 


126  ALGEBRA. 

2^    __a b__ _  b^  —  a^      nn     (2 x  —  3 vrif  _ x  —  3m 

-""'"      '    *  ■     {2x-Sny~  x-Sn' 


a 

% 

b 

b'-a' 

X  —  l 

x-b 

W-bx 

ax 

-  + 

bx 

=  a4-b. 

21.   _ii!l_  +  _i^  =  a  +  6.        23.    2(a+&)^x4:5_^--a^ 

x-{-b     x-{-  a  X  x—b     x-\-a 

24        1       ,      1  2x  —  a—b 


X  —  a     x  —  b     x{x  —  a  —  b) 

^•  +  4a4-6      4:X-^a-\-2 
X  -\-  a  +  b         x-\-a  —  b 


gg     g;  +  4  g  +  ^  _^  4:X-^ai-2b  ^  ^ 


SOLUTION  OF  EQUATIONS  INVOLVING  DECIMALS. 
154.   1.    Solve  the  equation  .17  a;  -  .23  =  .113  x  +  .112. 

Transposing,  .17  x  -  .llSx  =  .23  +  .112. 

Uniting  terms,  .057  x  =  .342. 

Dividing  by  .057,  x  =  '^^  =  6,  Ans. 

.057 

EXAMPLES. 
Solve  the  following  equations  : 

2.  2.9  a;  -  1.98  =  1.4  a;  -  1.845. 

3.  .05a;  +  .117  =  .186cc-.2a;-.139. 

4.  .6  a; -.265 +  .03  =  .4 +  .66  a; -.187  a;. 

5.  .4(1.7aj-.6)  =  .95x  +  5.16. 

6.  .08(35  aj  -  2.3)  =  . 9(7  a5  +  . 18) -.997. 

7.  2.8.-'^^^+ -^^^^  =  .5. -.064. 


8    3  39      .4  a; +  .708  ^18      .3 
2a;  5       x 

a    .7  a; +  .371      .3a; -.256^  ^^ 


10. 


.9  .6 

2-3a;      3a;-14     a;-2      10a;-9 


1.5  9  1.8  2.25 


3     7^^    ^^ 
Q  SIMPLE   EQUATIONS.  127 

PROBLEMS. 

155.  1.  Divide  43  into  two  parts  such  that  three-eighths 
of  one  part  may  equal  two-ninths  of  the  other. 

Let  X  =  one  part. 

Then,  43  —  x  =  the  other. 

By  the  conditions,  ^  =  -  (43  -  x) . 

J  8       9 

Clearing  of  fractions,  27  x  =  16  x  43  -  16  x. 

Transposing,  43  x  =  16  x  43. 

Dividing  by  43,  x  =  16,  one  part. 

Whence,  43  —  x  =  27,  the  other  part. 

2.  The  fifth  part  of  a  number  exceeds  its  eighth  part  by 
3 ;  what  is  the  number  ? 

3.  What  number  is  that  from  which  if  four-sevenths  of 
itself  be  subtracted,  the  result  will  equal  three-fourths  of 
the  number  diminished  by  18  ? 

4.  What  number  exceeds  the  sum  of  its  third,  sixth,  and 
fourteenth  parts  by  18  ? 

5.  Divide  45  into  two  parts  such  that  the  sum  of  four- 
ninths  the  greater  and  two-thirds  the  less  shall  equal  24. 

6.  Divide  56  into  two  parts  such  that  five-eighths  the 
greater  shall  exceed  seven-twelfths  the  less  by  6. 

7.  Divide  $  124  between  A,  B,  and  C  so  that  A's  share 
may  be  five-sixths  of  B's,  and  C's  nine-tenths  of  A's. 

8.  A  man  travelled  768  miles.  He  went  four-fifths  as 
many  miles  by  water  as  by  rail,  and  five-twelfths  as  many 
by  carriage  as  by  water.  How  many  miles  did  he  travel  in 
each  manner  ? 

9.  A's  age  is  three-eighths  of  B's,  and  eight  years  ago 
it  was  two-sevenths  of  B's  age ;  find  their  ages  at  present. 


128  ALGEBRA. 

10.  A  has  $  52,  and  B  $  38.  After  giving  B  a  certain 
sum,  A  has  only  three-sevenths  as  much  money  as  B.  What 
sum  was  given  to  B  ? 

11.  I  paid  a  certain  sum  for  a  picture,  and  the  same  price 
for  a  frame.  If  the  picture  had  cost  $  4  more,  and  the 
frame  30  cents  less,  the  price  of  the  frame  would  have  been 
one-third  that  of  the  picture.     Find  the  cost  of  the  picture. 

12.  A  can  do  a  piece  of  work  in  8  days  which  B  can  per- 
form in  10  days.  In  how  many  days  can  it  be  done  by  both 
working  together  ? 

Let  X  =  the  number  of  days  required. 

Then,  -  =  the  part  both  can  do  in  one  day. 

Also,  -  =  the  part  A  can  do  in  one  day, 

8 

and  —  =  the  part  B  can  do  in  one  day. 

10 

By  the  conditions,  -  H —  =  — 

8     10     « 

5  ic  +  4  ic  =  40. 

9x  =  40. 

Whence,  x  =  4|,  the  number  of  days  required. 

13.  The  second  digit  of  a  number  exceeds  the  first  by  2 ; 
and  if  the  number,  increased  by  6,  be  divided  by  the  sum  of 
its  digits,  the  quotient  is  5.     Find  the  number. 

Let  X  =  the  first  digit. 

Then,  x  -\- 2  =  the  second  digit, 

and  2  X  +  2  =  the  sum  of  the  digits. 

The  number  itself  is  equal  to  10  times  the  first  digit,  plus  the  second, 
Then,        lOx  +  (ic  +  2),  or  11  x  +  2  =  the  number. 

By  the  conditions,   li^-+l±^  =  5. 

2x  +  2 

llx  +  8  =  10x  + 10. 
Whence,  x  =  2. 

Then,  11  x  4-  2  =  24,  the  number  required. 


SIMPLE   EQUATIONS.  129 

14.  A  can  do  a  piece  of  work  in  18  days,  and  B  can  do 
the  same  in  24  days.  In  how  many  days  can  it  be  done  by 
both  working  together  ? 

15.  A  can  do  a  piece  of  work  in  3^  hours  which  B  can  do 
in  3|  hours,  and  C  in  3|  hours.  In  how  many  hours  can 
it  be  done  by  all  working  together  ? 

16.  A  tank  can  be  filled  by  one  pipe  in  9  hours,  and 
emptied  by  another  in  21  hours.  In  what  time  will  the 
tafik  be  filled  if  both  pipes  be  opened  ? 

17.  A  vessel  can  be  filled  by  three  taps;  by  the  first 
alone  in  7|  minutes,  by  the  second  alone  in  4^  minutes,  and 
by  the  third  alone  in  4|  minutes.  In  what  time  will  it  be 
filled  if  all  the  taps  be  opened  ? 

18.  The  first  digit  of  a  number  is  4  less  than  the  second ; 
and  if  the  number  be  divided  by  the  sum  of  its  digits,  the 
quotient  is  4.     Find  the  number. 

19.  The  second  digit  of  a  number  is  one-fourth  of  Ihe 
first;  and  if  the  number,  diminished  by  10,  be  divided  by 
the  difference  of  its  digits,  the  quotient  is  12.  Find  the 
number. 

20.  If  a  certain  number  be  diminished  by  23,  one-fourth 
of  the  result  is  as  much  below  37  as  the  number  itself  is 
above  56.     Find  the  number. 

21.  What  number  is  that,  seven-eighths  of  which  is  as 
much  below  21  as  three-tenths  of  it  exceeds  2\  ? 

22.  B  is  24  years  older  than  A ;  and  when  A  is  twice  his 
present  age,  B  will  be  f  as  old  as  he  now  is.  How  old  is 
each? 

23.  The  denominator  of  a  fraction  exceeds  the  numerator 
by  5.  If  the  denominator  be  decreased  by  20,  the  resulting 
fraction,  increased  by  1,  is  equal  to  twice  the  original  frac- 
tion.    Find  the  fraction. 


130  ALGEBRA. 

24.  Divide  44  into  two  parts  such  that  one  divided  by 
the  other  shall  give  2  as  a  quotient  and  5  as  a  remainder. 

Let  X  =  the  divisor. 

Then,  44  —  x  =  the  dividend. 

Now  since  the  dividend  is  equal  to  the  product  of  the  divisor  and 
quotient,  plus  the  remainder,  we  have 

44  -  X  =  2  cc  +  5. 

-3cc  =  -39. 

Whence,  x  =  13,  the  divisor, 

and  44  —  X  =  31,  the  dividend. 

25.  Two  persons,  A  and  B,  63  miles  apart,  start  at  the 
same  time  and  travel  towards  each  other.  A  travels  at  the 
rate  of  4  miles  an  hour,  and  B  at  the  rate  of  3  miles  an 
hour.     How  far  will  each  have  travelled  when  they  meet  ? 

Let  4x  =  the  number  of  miles  that  A  travels. 

Then,  Sx  =  the  number  of  miles  that  B  travels. 

By  the  conditions, 

4ic  +  3x  =  63. 

7x  =  6S. 

x  =  9. 

Whence,  4  a;  =  36,  the  number  of  miles  that  A  travels, 

and  3x  =  27,  the  number  of  miles  that  B  travels. 

Note.  It  is  often  advantageous,  as  in  Ex.  25,  to  represent  the 
unknown  quantity  by  some  multiple  of  x  instead  of  by  x  itself. 

26.  Divide  49  into  two  parts  such  that  one  divided  by 
the  other  may  give  2  as  a  quotient  and  7  as  a  remainder. 

27.  Two  men,  A  and  B,  66  miles  apart,  set  out  at  the 
same  time  and  travel  towards  each  other.  A  travels  at  the 
rate  of  15  miles  in  4  hours,  and  B  at  the  rate  of  9  miles 
in  2  hours.  How  far  will  each  have  travelled  when  they 
meet? 


SIMPLE  EQUATIONS.  131 

28.  Divide  134  into  two  parts  such  that  one  divided  by 
the  other  may  give  3  as  a  quotient  and  26  as  a  remainder. 

29.  The  denominator  of  a  fraction  is  7  less  than  the 
numerator ;  and  if  5  be  added  to  the  numerator,  the  value 
of  the  fraction  is  f.     Find  the  fraction. 

30.  The  second  digit  of  a  number  exceeds  the  first  by  4 ; 
and  if  the  number,  increased  by  39,  be  divided  by  the  sum 
of  its  digits,  the  quotient  is  7.     Find  the  number. 

31.  I  paid  a  certain  sum  for  a  horse,  and  seven-tenths  as 
much  for  a  carriage.  If  the  horse  had  cost  $  70  less,  and 
the  carriage  $  50  more,  the  price  of  the  horse  would  have 
been  four-fifths  that  of  the  carriage.  What  was  the  cost  of 
each? 

32.  A  can  do  a  piece  of  work  in  15  hours,  which  B  can 
do  in  25  hours.  After  A  has  worked  for  a  certain  time,  B 
completes  the  job,  working  9  hours  longer  than  A.  How 
many  hours  did  A  work  ? 

33.  A  man  owns  a  horse,  a  carriage  worth  $100  more 
than  the  horse,  and  a  harness.  The  horse  and  harness  are 
together  worth  three-fourths  the  value  of  the  carriage,  and 
the  carriage  and  harness  are  together  worth  $  50  less  than 
twice  the  value  of  the  horse.     Find  the  value  of  each. 

34.  The  rate  of  an  express  train  is  |  that  of  a  slow  train, 
and  it  covers  180  miles  in  two  hours'  less  time  than  the 
slow  train.     Find  the  rate  of  each  train. 

35.  Two  men,  A  and  B,  57  miles  apart,  set  out,  B  20 
minutes  after  A,  and  travel  towards  each  other.  A  travels 
at  the  rate  of  6  miles  an  hour,  and  B  at  the  rate  of  5 
miles  an  hour.  How  far  will  each  have  travelled  when 
they  meet? 

36.  A  grocer  buys  eggs  at  the  rate  of  4  for  7  cents.  He 
sells  one-fourth  of  them  at  the  rate  of  5  for  12  cents,  and 
the  remainder  at  the  rate  of  6  for  11  cents,  and  makes  27 
cents  by  the  transaction.     How  many  eggs  did  he  buy  ? 


132  ALGEBHA. 

37.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  a  watch  opposite  to  each  other  ? 

Let  X  =  the  number  of  minute-spaces  passed  over  by  the  minute- 
hand  from  3  o'clock  to  the  required  time. 

Then,  since  the  hour-hand  is  15  minute-spaces  in  advance  of  the 
minute-hand  at  3  o'clock,  x  —  15  —  30,  or  x  —  45,  will  represent  the 
number  of  minute-spaces  passed  over  by  the  hour-hand. 

But  the  minute-hand  moves  12  times  as  fast  as  the  hour-hand. 

Whence,  x  =  12(x  — 45). 

x  =  12X-540. 
-llx  =  -540. 

X  =  49j-V. 
Then  the  required  time  is  49yx  minutes  after  3  o'clock. 

38.  At  what  time  between  1  and  2  are  the  hands  of  a 
watch  opposite  to  each  other  ? 

39.  At  what  time  between  6  and  7  is  the  minute-hand  of 
a  watch  5  minutes  in  advance  of  the  hour-hand  ? 

40.  At  what  time  between  4  and  5  are  the  hands  of  a 
watch  together? 

41.  At  what  time  between  5  and  5.30  are  the  hands  of  a 
watch  at  right  angles  to  each  other  ? 

42.  The  sum  of  the  digits  of  a  number  is  15 ;  and  if  the 
number  be  divided  by  its  second  digit,  the  quotient  is  12, 
and  the  remainder  3.     Find  the  number. 

43.  A  man  has  11  hours  at  his  disposal.  How  far  can 
he  ride  in  a  coach  which  travels  4|-  miles  an  hour,  so  as  to 
return  in  time,  walking  back  at  the  rate  of  3|  miles  an  hour  ? 

44.  A,  B,  and  C  together  can  do  a  piece  of  work  in  If 
days ;  B's  work  is  one-half  of  A's,  and  C's  three-fourths  of 
B's.    How  many  days  will  it  take  each  working  alone  ? 

45.  At  what  time  between  9  and  10  are  the  hands  of  a 
watch  together  ? 


SIMPLE   EQUATIONS.  133 

46.  A,  B,  C,  and  D  found  a  sum  of  money.  They  agreed 
that  A  should  receive  $  4  less  than  one-third,  B  $  2  more 
than  one-fourth,  C  $3  more  than  one-iifth,  and  D  the 
remainder,  $  25.     How  much  did  A,  B,  and  C  receive  ? 

47.  At  what  time  between  8  and  9  are  the  hands  of  a 
watch  opposite  to  each  other  ? 

48.  A  vessel  can  be  emptied  by  three  taps ;  by  the  first 
alone  in  90  minutes,  by  the  second  alone  in  144  minutes, 
and  by  the  third  alone  in  4  hours.  In  what  time  will  it  be 
emptied  if  all  the  taps  be  opened  ? 

49.  A  and  B  start  in  business,  B  putting  in  f  as  much 
capital  as  A.  The  first  year,  A  loses  $  500,  and  B  gains  ^ 
of  his  money;  the  second  year,  A  gains  ^  of  his  money, 
and  B  loses  $  205 ;  and  they  have  now  equal  amounts. 
How  much  had  each  at  first  ? 

50.  A  man  buys  two  pieces  of  cloth,  one  of  which  con- 
tains 6  yards  more  than  the  other.  For  the  larger  he  pays 
at  the  rate  of  $  7  for  10  yards,  and  for  the  smaller  at  the 
rate  of  $  5  for  3  yards.  He  sells  the  whole  at  the  rate  of 
9  yards  for  $  11,  and  makes  $  5  on  the  transaction.  How 
many  yards  were  there  in  each  piece  ? 

51.  A  man  loaned  a  certain  sum  for  3  years  at  5  per  cent 
compound  interest;  that  is,  at  the  end  of  each  year  there 
was  added  ^  to  the  sum  due.  At  the  end  of  the  third 
year,  there  was  due  him  $2130.03.  Find  the  amount 
loaned. 

62.  At  what  times  between  7  and  8  are  the  hands  of  a 
watch  at  right  angles  to  each  other  ? 

53.  At  what  time  between  2  and  3  is  the  hour-hand  of  a 
watch  one  minute  in  advance  of  the  minute-hand  ? 

54.  Grold  is  19}  times  as  heavy  as  water,  and  silver  10|- 
times.  A  mixed  mass  weighs  1960  oz.,  and  displaces  120  oz. 
of  water.     How  many  ounces  of  each  metal  does  it  contain  ? 


134  ALGEBRA. 

55.  A  merchant  increases  Ms  capital  annually  by  one- 
third  of  it,  and  at  the  end  of  each  year  takes  out  $  1800 
for  expenses.  At  the  end  of  three  years,  after  taking  out 
his  expenses,  he  finds  that  his  capital  is  $  3800.  What  was 
his  capital  at  first  ? 

56.  A  and  B  together  can  do  a  piece  of  work  in  2|  days, 
B  and  C  in  2j§j-  days,  and  C  and  A  in  2|-  days.  How  many 
days  will  it  take  each  working  alone  ? 

57.  A  alone  can  do  a  piece  of  work  in  15  hours ;  A  and 
B  together  can  do  it  in  9  hours,  and  A  and  C  together  in  10 
hours.  A  commences  work  at  6  a.m.  ;  at  what  hour  can  he 
be  relieved  by  B  and  0,  so  that  the  work  may  be  completed 
at  8  P.M.  ? 

58.  A  man  invests  j^  of  a  certain  sum  in  4J  per  cent 
bonds,  and  the  balance  in  3^  per  cent  bonds,  and  finds  his 
annual  income  to  be  f  117.50.  How  much  does  he  invest 
in  each  kind  of  bond  ? 


The  annual  income  from  p  dollars,  invested  at  r  per  cent,  is  rep- 
resented by  ^^.  \ 

59.  An  express  train  whose  rate  is  36  miles  an  hour  starts 
54  minutes  after  a  slow  train,  and  overtakes  it  in  1  hour 
48  minutes.     What  is  the  rate  of  the  slow  train  ? 

60.  At  what  time  between  10  and  11  is  the  minute-hand 
of  a  watch  25  minutes  in  advance  of  the  hour-hand  ? 

61.  A  woman  sells  half  an  egg  more  than  half  her  eggs. 
She  then  sells  half  an  egg  more  than  half  her  remaining 
eggs.  A  third  time  she  does  the  same,  and  now  she  has 
sold  all  her  eggs.     How  many  had  she  at  first  ? 

62.  A  man  invests  two-fifths  of  his  money  in  6 J  per  cent 
bonds,  two-ninths  in  5^  per  cent  bonds,  and  the  balance  in 
3|  per  cent  bonds.  His  income  from  the  investments  is 
f  915.     Find  the  amount  of  his  property. 


SIMPLE   EQUATIONS.  135 

63.  A  man  starts  in  business  with  $  8000,  and  adds  to 
his  capital  annually  one-fourth  of  it.  At  the  end  of  each 
year  he  sets  aside  a  fixed  sum  for  expenses.  At  the  end  of 
three  years,  after  deducting  the  fixed  sum  for  expenses, 
his  capital  is  reduced  to  $  6475.  What  are  his  annual 
expenses  ? 

64.  If  19  oz.  of  gold  weigh  18  oz.  in  water,  and  10  oz.  of 
silver  weigh  9  oz.  in  water,  how  many  ounces  of  each  metal 
are  there  in  a  mixed  mass  weighing  127  oz.  in  air,  and 
117  oz.  in  water  ? 

65.  A  fox  is  pursued  by  a  hound,  and  has  a  start  of  63 
of  her  own  leaps.  The  fox  makes  4  leaps  while  the  hound 
makes  3 ;  but  the  hound  in  5  leaps  goes  as  far  as  the  fox 
in  9.  How  many  leaps  does  each  make  before  the  hound 
catches  the  fox  ? 

(Let  4  a;  =  the  number  of  leaps  made  by  the  fox,  and  3  a;  =  the 
number  made  by  the  hound.) 

66.  A  merchant  increases  his  capital  annually  by  one- 
third  of  it,  and  at  the  end  of  each  year  sets  aside  $  2700  for 
expenses.  At  the  end  of  three  years,  after  deducting  the 
sum  for  expenses,  he  has  ||  of  his  original  capital.  Find 
his  original  capital. 

PROBLEMS  INVOLVING  LITERAL  EQUATIONS. 

156.  1.  Divide  a  into  two  parts  such  that  m  times  the 
first  shall  exceed  n  times  the  second  by  b. 

Let  X  =  one  part. 

Then,  a  —  x  =  the  other  part. 

By  the  conditions,  inx  =  n(a  -x)  +  b. 

mx  =  an  —  nx  -{■  b. 

mx  -{-  nx  =  an  +  b. 

x(jtn  +  w)  =  an  +  b. 


136  ALGEBRA. 

Whence,  x  —  ^^  "^    ,  one  part, 

m  +  n 

and  ^  _  ^  ^  ^  _an  +  b  ^am  +  an  -  an  -  b 


m  +  n  m  +  n 

am  —  b 


m  +  n 


the  other  part. 


Note.  The  results  can  be  used  as  formulae  for  solving  any  prob- 
lem of  the  above  form. 

Thus,  let  it  be  required  to  divide  25  into  two  parts  such  that  4  times 
the  first  shall  exceed  3  times  the  second  by  37. 
Here,  a  =  25,  m  =  4,  w  =  3,  and  6  =  37. 
Substituting  these  values  in  the  results  of  Ex.  1 , 

thefirstpart  ^25x3  +  37^75  +  37^112^ 

7  7  7- 

and  the  second  part  =  26x^^l37  ^  10^-37  ^  63  ^  , 

7  7  7 

2.  Divide  a  into  two  parts  such  that  m  times  the  first 
shall  equal  n  times  the  second. 

3.  A  is  m  times  as  old  as  B,  and  a  years  ago  he  was 
n  times  as  old.     Find  their  ages  at  present. 

4.  A  can  do  a  piece  of  work  in  m  hours,  which  B  can 
do  in  n  hours.  In  how  many  hours  can*  it  be  done  by  both 
working  together  ? 

5.  A  vessel  can  be  filled  by  three  taps ;  by  the  first 
alone  in  a  minutes,  by  the  second  alone  in  b  minutes,  and 
by  the  third  alone  in  c  minutes.  In  how  many  minutes 
will  it  be  filled  if  all  the  taps  be  opened  ? 

6.  A  has  m  dollars,  and  B  has  n  dollars.  After  giving 
A  a  certain  sum,  B  has  r  times  as  much  money  as  A.  What 
sum  was  given  to  A  ? 

7.  A  gentleman  distributing  some  money  among  beggars, 
found  that  in  order  to  give  them  a  cents  each,  he  would  need 
b  cents  more.  He  therefore  gave  them  c  cents  each,  and  had 
d  cents  left.     How  many  beggars  were  there  ? 


SIMPLE   EQUATIONS.  137 

8.  A  man  has  a  hours  at  his  disposal.  How  far  can  he 
ride  in  a  coach  which  travels  h  miles  an  hour,  so  as  to  return 
home  in  time,  walking  back  at  the  rate  of  c  miles  an  hour  ? 

9.  A  courier  who  travels  a  miles  in  a  day  is  followed 
after  n  days  by  another  who  travels  h  miles  in  a  day.  In 
how  many  days  will  the  second  overtake  the  first  ? 

10.  What  principal  at  r  per  cent  interest  will  amount  to 
a  dollars  in  t  years  ? 

11.  In  how  many  years  will  p  dollars  ainount  to  a  dollars 
at  r  per  cent  interest  ? 

12.  At  what  rate  per  cent  will  p  dollars  amount  to  a  dol- 
lars in  t  years  ? 

13.  Divide  a  into  two  parts,  such  that  one  divided  by  the 
other  may  give  6  as  a  quotient  and  c  as  a  remainder. 

14.  Two  men,  A  and  B,  a  miles  apart,  start  at  the  same 
time,  and  travel  towards  each  other.  A  travels  at  the  rate 
of  m  miles  an  hour,  and  B  at  the  rate  of  n  miles  an  hour. 
How  far  will  each  have  travelled  when  they  meet  ? 

15.  A  grocer  mixes  a  pounds  of  coffee  worth  m  cents  a 
pound,  h  pounds  worth  n  cents  a  pound,  and  c  pounds  worth 
p  cents  a  pound.     Find  the  cost  per  pound  of  the  mixture. 

16.  A  banker  has  two  kinds  of  money.  It  takes  a  pieces 
of  the  first  kind  to  make  a  dollar,  and  h  pieces  of  the  second 
kind.  If  he  is  offered  a  dollar  for  c  pieces,  how  many  of 
each  kind  must  he  give  ? 

17.  Divide  a  into  three  parts,  such  that  the  first  may  be 
m  times  the  second,  and  the  second  n  times  the  third. 

18.  A  and  B  together  can  do  a  piece  of  work  in  m  hours, 
B  and  C  in  n  hours,  and  C  and  A  inp  hours.  In  how  many 
hours  can  each  alone  do  the  work  ? 


X38  ALGEBRA. 

XIV.    SIMULTANEOUS  EQUATIONS. 

CONTAINING  TWO  UNKNOWN  QUANTITIES. 

157.  If  a  rational  and  integral  monomial  (§  69)  involves 
two  or  more  letters,  its  degree  ivith  respect  to  them  is  denoted 
by  the  sum  of  their  exponents. 

Thus,  2  a^b^xy^  *is  of  the  fourth  degree  with  respect  to 
X  and  y. 

158.  If  each  term  of  an  equation  containing  one  or  more 
unknown  quantities  is  rational  and  integral,  the  degree  of 
the  equation  is  the  degree  of  its  term  of  highest  degree. 

Thus,  if  X  and  y  represent  unknown  quantities, 

ax  —  by  =  c  is  an  equation  of  the  first  degree. 

ic^  +  4  ic  =  —  2  is  an  equation  of  the  second  degree. 

2  x^  —  Sxy^  =  5  is  an  equation  of  the  third  degree ;  for  the 
term  3  xy"^  is  the  term  of  highest  degree,  and  3  xy^  is  of  the 
third  degree. 

159.  An  equation  containing  two  or  more  unknown  quan- 
tities is  satisfied  by  an  indefinitely  great  number  of  sets  of 
value  of  these  quantities. 

Consider,  for  example,  the  equation  x  -\-y  =  5. 

If  a;  =  1,  we  have  1  -\-y  =  5,  or  ?/  =  4. 

If  a;  =  2,  we  have  2  -\-y  =  5,  ov  y  =  3;  and  so  on. 

Thus  the  equation  is  satisfied  by  any  one  of  the  sets  of 

values 

x  =  l,     y  =  4:; 

x=:2,     y  =  3;  etc. 

For  this  reason,  an  equation  containing  two  or  more  un- 
known quantities  is  called  an  indetermmate  equation. 


SIMULTANEOUS  EQUATIONS.  139 

160.  Two  equations,  each  containing  two  unknown  quan- 
titieSj  are  said  to  be  Independent  when  one  of  them  is  satis- 
fied by  sets  of  values  of  the  unknown  quantities  which  do 
not  satisfy  the  other. 

Consider,  for  example,  the  equations  x-\-y  =  5,  x  —  y  =  3. 

The  first  equation  is  satisfied  by  the  set  of  values 
x  =  3.  y  =  2,  which  does  not  satisfy  the  second. 

Therefore,  the  equations  are  independent. 

But  the  equations  x  -\-y  =  5,  2  x  -\-  2  y  =  10,  are  not  inde- 
pendent; for  the  second  equation  can  be  reduced  to  the 
form  of  the  first  by  dividing  each  term  by  2;  and  hence 
every  set  of  values  of  x  and  y  which  satisfies  one  equation 
will  also  satisfy  the  other. 

161.  Let  there  be  two  independent  equations  (§  160), 
each  of  the  first  degree,  containing  the  unknown  quantities 
X  and  y,  SiS  x  +  y  =  5,  x  —  y  =  3. 

By  §  159,  each  equation  considered  by  itself  is  satisfied 
by  an  indefinitely  great  number  of  sets  of  values  of  x  and  y. 

But  there  is  only  one  set  of  values  of  x  and  y,  i.e.,  a;  =  4, 
y  =  1,  which  satisfies  both  equations  at  the  same  time. 

A  series  of  equations  is  called  Simultaneous  when  each 
contains  two  or  more  unknown  quantities,  and  every  equa- 
tion of  the  series  is  satisfied  by  the  same  set  of  values  of 
the  unknown  quantities. 

162.  To  solve  a  series  of  simultaneous  equations  is  to  find 
the  set  of  values  of  the  unknown  quantities  involved  which 
satisfies  all  the  equations  at  the  same  time. 

163.  Two  independent,  simultaneous  equations  may  be 
solved  by  combining  them  in  such  a  way  as  to  form  a  single 
equation  containing  but  one  unknown  quantity. 

This  operation  is  called  Elimination. 

There  are  three  principal  methods  of  elimination. 


140 


4( 

ALGEBRA. 


1.    Solve  the  equations 


164.  I.   Elimination  by  Addition  or  Subtraction. 

5x-3y  =  19. 

7x-\-4:y=    2. 

Multiplying  (1)  by  4,  20  x  -  12  y  =    76. 

Multiplying  (2)  by  3,  21  x  +  12  ?/  =      6. 

Adding  (3)  and  (4),  41  x  =    82. 

Whence,  z=     2. 

Substituting  the  value  of  x  in  (1),  10  —  Sy  =    19. 

-.Sy=     9. 
Whence,  y  =  —  S. 

The  above  is  an  example  of  elimination  hj» addition. 

1. 


(1 

(2) 
(3) 
(4) 

u 


4    1> 


2.    Solve  the  equations 

Multiplying  (1)  by  2, 
Multiplying  (2)  by  3, 

Subtracting  (4)  from  (3), 
Whence, 


15x-\-Sy 

y 


llOa^-7 


-24. 


30x+  16?/  = 
30x-21z/  = 


2. 

72. 


(1) 
(2) 

(3) 
(4) 


37  ?/  =      74. 

?/=        2. 


Substituting  the  value  of  y  in  (2),  10 x  —  14  =  —  24. 

10x  =  -  10. 
Whence,  •  x=:—    1. 

The  abote  is  an  example  of  elimination  by  subtraction. 

Rule. 

If  necessary,  multiply  the  given  equations  by  such  numbers 
as  will  make  the  coefficients  of  one  of  the  unknown  quantities 
in  the  resulting  equations  of  equal  absolute  value. 

Add  or  subtract  the  resulting  equations  according  as  the 
coefficients  of  equal  absolute  value  are  of  unlike  or  like  sign. 

Note.  If  the  coefficients  which  are  to  be  made  of  equal  absolute 
value  are  prime  to  each  other,  each  may  be  used  as  the  multiplier  for 
the  other  equation ;  but  if  they  are  not  prime  to  each  other,  such 
multipliers  should  be  used  as  will  produce  their  lowest  common  mul- 
tiple. Thus,  in  Ex.  1,  to  make  the  coefficients  of  y  of  equal  absolute 
value,  we  multiplied  (1)  by  4  and  (2)  by  3  ;  but  in  Ex.  2,  to  make  the 
coefficients  of  a  3f  equal  absolute  value,  since  the  L.  C.  M.  of  10  and 
15  is  30,  we  multiplied  (1)  by  2  and  (2)  by  3. 


SIMULTANEOUS  EQUATIONS. 


141 


SX+'+ii.^'^'^-^     EXAMPLES. 
Solve  by  the  method  of  addition  or  subtraction 


■■( 


10. 


5x-\-   4^=22. 

3a;  +      y  =  9. 
X-    6y  =  -10. 

2x-    7^  =  -15. 

7x-    2y  =  Sl. 

4a; -h   3y  =  -S. 

6x-^lly  =  -2S. 

5y-lSx  =  S. 

6x-\-    2y  =  -S. 

5x-    3y  =  -6. 

4:X-\-15y  =  7. 
14a; -f    6^  =  9. 
12a;-    5y  =  10. 
30a;+lly  =  -69. 

3a;H-    7y  =  2. 

7x-{-    Hy  =  -2. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


rl7a; 
113  a; 


17a;-|-102/  =  -30. 
35  2/ =  -40. 
rlla;—   5y=    4. 
1    9a;  4-    6^  =  10. 
Sx-\-    9y  =  -4:. 
Sy-    9a;  =  77. 
5x—    9y  =  1. 
Sx-10y  =  -5. 
i21x-    Sy  =  92. 
[    9a;  +  17y  =  19. 
10a;-ll2/  =  -27. 
10?/ -11  a;  =  36. 
^22x-\-Wy  =  9. 
.18a; +  252/ =  71. 
r    5a;-242/  =  -123. 
119a; -36?/ =  -81. 


165.  II.   Elimination  by  Substitution. 

(7x-9y 


1.    Solve  the  equations 

Transposing  —  6x  in  (2), 
Whence, 


Sy- 

y 


15. 
5a;  =  -17. 

5X-17. 
5X-17 


'  "8 

Substituting  this  in  (1),  7x-  q/^^-^^N  ^  ^^ 


Clearing  of  fractions, 
Expanding, 


Whence, 

Substituting  the  value  of  x  in  (3) 


66x-9(5x-17)=120. 
56x-45x  +  163  =  120. 
llx  =  -33. 
x  =  -3. 
15- 


y  = 


17 


8 


-4. 


\y 


I 

(1) 

(2) 
(3) 


0 


n 


.'?<^r  invf'V 


142 


■^ 


ALGEBRA. 
Rule. 


From  one  of  the  given  equations  find  the  value  of  one  of  the 
unknown  quantities  in  terms  of  the  other,  and  substitute  this 
value  in  place  of  that  quantity  in  the  other  equation. 


Ml 


3 


EXAMPLES. 

Solve  by  the  method  of  substitution : 
^x+    2y  =  17.  _     r    8a; 

x-\-      y  =  16. 

X—    6y  =  2. 
3y-    8aj  =  29. 
2x-    3?/  =  -14. 
3x-\-    7y  =  4S. 
Sx-\-    5y=:5. 
3x-    2y  =  29. 
(    2x+    5y  =  13. 
Ix-    4j/  =  -19. 
3x+    7y  =  -23. 
5x-\-    Ay  =  -23. 
5x+    9y  =  8. 
6y-    9a;  =  -7. 
5x+    Sy  =  -6. 
10x-12v  =  -5. 


e-\ 


10. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


r   ««—   32/  =  — 6. 
I    ix-{-^d.y  =  \.i 

I 


7x^-  'sV^-io.  '- 

llla;+    6?/ =  -19. 
6  X  —  10  y  =  5. 
15  2/  —  14  ic  =  —  15. 
(    9a;+    Sy  =  -6. 
[l2x-{-10y  =  -7. 
16x  —  lly  =  o6. 
12  a;-    7y  =  37. 
7x-    82/  =  -43. 
5y-    6a;  =  35. 
6x-    9y  =  19. 
15  a;  +    7  2/  =  —  41. 
5  a;-    8  2/ =  60. 
ex-\-    7y  =  -ll. 


166.  III.   Elimination  by  Comparison. 


y  - 


1.    Solve  the  equations 

f  2a; -52/ =  -16. 
3a;  +  72/  =  5. 

(1) 
(2) 

Transposing  -  5?/  in  (1), 

2x  =  5y-16. 

Whence, 

2 

(3) 

Transposing  7  ?/  in  (2), 

3x  =  6-72/. 

SIMULTANEOUS  EQUATIONS. 


143 


Whence, 

Equating  the  values  of  x, 

Clearing  of  fractions, 

Whence, 

Substituting  the  value  of  y  in  (3) 
Rule 


7y 


3 

5_y_ 

-16 

5-' 

Ll. 

2 

3 

15?/ 

-48  = 

=  10- 

Uy. 

29y^ 

=  58. 

y-- 

=  2. 

X  - 

10- 

J6_ 

3. 


From  each  of  the  given  equations  find  the  value  of  the  same 
unknown  quantity  in  terms  of  the  other,  and  pkice  these  values 
equal,  to  each  other. 

EXAMPLES. 


Solve  by  the  method  of  comparison : 


I 


2a; -f      y  =  9. 

OX+  3  2/ =  25. 

x-^  2y  =  -2. 

4.x-  7y  =  37. 

(    6x—  oy  =  — 10. 

1    ox-  2y  =  -17. 

rll.^•+  42/ =  3. 

1    8a;+  9y  =  -10. 

7X+  32/  =  -9. 

6y-  9.x- =  28. 
12x-25y  =  l. 


10. 


llOa; 


12. 


13 


fl2a;-   6  7/ =19. 
4«/-    3a;^-ll. 
Qx—    7  y  =  —  12. 
92/ =  -12. 
{Wx-\-    Sy  =  -U. 
6x-\-12y  =  l. 
5x-\-    3y  =  27. 
3a;  =  -26. 


r    5a;H- 
*  1    82/- 
^^    r   2x-\-   5y  =  -27 
'   [nx-\-    62/  =  -41 


(rZx  —  -Zby  =  l.  --     f    oa;—    y?/  =  (5. 

1    4:X-j-10y  =  -7.  '  I    7x-\-    42/  =  29. 


9. 


6  a;  —    5y  =  1. 
9a; +  10?/ =  12. 
3a;-    Sy  =  -17. 
7a;+    62/  =  -15. 


16. 


17. 


10a;-f  182/  =  -ll 
.  14  2/  —  15  a;  =  —  4. 
9a;_    72/  =  -85. 
4  a;  —  11 2/  =  —  93. 


144 


ALGEBRA. 


MISCELLANEOUS  EXAMPLES. 

167.  iB^fore  applying  either  method  of  elimination,  each 
of  the  given  equations  should  be  reduced  to  its  simplest 
form.- 

r_7         3 

1.    Solve  the  equations 


0. 


(1) 


ajH-3     y  +  4. 
x(y-2)-y(x-5)=-lS.    (2) 


From  (1),  7i/  + 28-3x-9  =  0,  or  1y  -Sx=-\9. 

From  (2),  xy  -  2x  -  xy  -{-  ^y  =  -  IS^  or    by  —  2x  =  —lS. 
Multiplying  (3)  by  2, 
Multiplying  (4)  by  3, 
Subtracting  (5)  from  (6), 
Substituting  the  value  of  y  in  (4), 

Whence,  • 

Solve  the  following : 

'2x     Sy^_7 
3        4  2* 

x_2y^n 
4       5        2' 


(3) 
(4) 
(6) 
(6) 


2.     -! 


4. 


(Sx-\-7y  =  12. 
x  +  2y  ,2x-\-y^^ 
4  3 

f  x-\-5y     2y  -\-x_     -. 
13  11     ~ 


6. 


7. 


2a;-3     2y-\-13 

6±x--y_     7 


1-x-y 
2x-\-Sy  = 


Sx-y 
8 


2. 


=  2. 


3-2a;     4  +  5y 
5  11 


4. 


(x  +  l)(y-h9)-(x  +  5)(y-7)  =  112. 
2x-\-3y-\-9  =  0, 


SIMULTANEOUS   EQUATIONS. 


10. 


2  3 

3  4 

X  —  y  _25     X  -\-y 

2     ~'~6  3 

x-\-y-9_y--x-6^Q 

2  3 


11. 


12. 


10a;-^^^=ll. 

7 

82,-^±^  =  _17. 


x-\-y-2^      1 
a;-^  3 

3.r  +  y-3^       1 
2y-;r  11*       i 


145      3 


13. 


14. 


15. 


16. 


17. 


18. 


^  +  8  =  20.-^. 

3^_2j^-3^^^ 
3  ^ 


x-^y     7  X  —  5y 


11 


+  3  =  0. 


?-?l=l 
5       7 


'8-y     2a;  +  3^y  +  3 
5  4  4    * 

"^i 3~-'^- 

3a5-|(^4aj  +  2/  +  |)  =  52/. 
Sx-^(2x-\-y-{-6)  =  5y. 
l(Sx  +  2y)-h2y-x)  =  -S. 


aj  +  22/  +  4 


~2a5-3(2/+i)1=0. 


1K^04<^^^) 


11 
lo' 


146  *  ALGEBRA, 


x-Sy     y-3x     q^^ 

2  2 


19. 


21. 


22. 


23. 


24. 


26. 


1     ,3  on     |.8a^  +  .052/  =  .6365. 

5        4^  7  \mx-Ay=.l. 

2         3^ 


7  ^  4 

7a^  +  3y  +  12^      3 
x~  5y  —  4: 

2     2x  +  Sy^  5x-2y  - 

3  ^  17 

3a;  +  4.^      ^—0 


3x-4.y     13 

5x-\-6  _lly  —  5  _^^ 
10        ~"^i     ~     ■ 
7  a;  _  55  y  —  12  _  orr 


a;-2      10 -a;     y-lO^^ 
5  3  4' 

2/ +  2     2a;  +  ?/     a;  +  13  _ 


0. 
6  32  16 


r2a;y-3     4y  +  5^^ 
25.  a;  +  2         a;  -  3         ^ 

[(2aj-32/  +  l)(3a;  +  ll2/)  + 252/^=  (3aj  +  8^)(2a;-2/> 


.322/-2.4^-:505l±2:6  =  _.8a.-:56^±^. 
^  .25  .5 

.07 a; +  .1  I  -Q^y  +  .l^Q 
.6        "^        .3 


SIMULTANEOUS  EQUATIONS. 


147 


168.  'Literal  Simultaneous  Equations. 

In  solving  literal  simultaneous  equations,  the  method  of 
elimination  by  addition  or  subtraction  is  usually  the  best. 


2. 


4. 


1. 

Solve  the  equations 

dx  -^  by  =  c, 
a'x  +  b'y  =  c'. 

(1) 
(2) 

Multiplying  (1)  by  6', 
Multiplying  (2)  by  6, 

ab'x  +  bb'y  =  b'c. 
a'bx  +  bb'y  =  be'. 

Subtracting, 
Whence, 

ab'x  -  a'bx  =  b'c  -  be'. 

^_b'c-  be' 

ab'  -  a'b 

Multiplying  (1)  by  a', 
Multiplying  (2)  by  a, 

aa'x  4-  a'by  =  a'c. 
aa'x  +  ab'y  =  ae'. 

(3) 

(4) 

Subtracting  (3)  from  (4), 

ab'y  -  a'by  =  ae'  —  a'e. 

Whence, 

ae'-a'c^ 
ab'  -  a'b 

EXAMPLES. 

Solve  the  following : 

(Sx-\-4:y  =  7a. 

7. 
8. 

.ab.     9- 

•|±M  =  „. 

1. 

ax  —  by  —  1. 
bx-{-ay  =  l. 

mx  4-  ny  =  p. 

bx-\-ay_      ^ 
m     n 

m'x-\-n'y  =p'. 

ax  -\-by  =  m, 
ex  —  dy  =  n. 

X       .y  _     2 
3m     6n         3 

a      b      c 

r  (a  —  b)x  —  by  =  a^  — 
\x-\-y  =  2a. 

^  +  2/^1. 
a'^b'     c' 

10.  (^«-*)- 

[ay  +  bx 

c  -  (a  -f  b) 
=  0. 

y  =  a'-^b'. 

148 
11. 


ALGEBRA. 


ax  —  by  =  2  ah. 
2bx-\-2ay  =  Sb^ 


19      (x-ay  =  b(l-{-ab). 
a\    ^^'     [b 


bx-\-y  =  a{l-\-  ab). 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


y 


ra^tb=^-''' 

x-y  =  2{a^-  62). 


(b  —  a)x  —  {a  —  c)  y  =  be  —  a^ 
(b  —  c)x  —  ay  =  —  ac. 


(b  -h  c)x  -\-  (b  —  c)  y  =  2  ab. 

(a  -\-  c) X  —  (a  —  c)y  =  2  ac. 

mx  -^ny  =  mn  (m^  -|-  n^. 

x-\-y  =  mn  (m  +  n). 

ax  — by  =  2  b. 


bx-\-ay 


^aPj-a^b_±ab^ 
ab 


(a  +  b)x—  (a  —  b)y  =  3  ab. 


r  (a  -f-  0)  ic  —  (a 
\(a  —  b)x—  (a 


(a  —  b)x—  (a  -\-b)y  =  ab. 

2x  —  b_  Sx  —  y 

a      ~  a-\-2b 
2x  —  b_a  —  2y 
a  b 


169.  Certain  equations  in  which  the  unknown  quantities 
occur  in  the  denominators  of  fractions  may  be  readily  solved 
without  previously  clearing  of  fractions. 


=  8. 


1.   Solve  the  equations 


fl0_9 
x      y 

X     y 


(1) 
(2) 


SIMULTANEOUS  EQUATIONS. 


149 


Multiplying  (1)  by  5, 
Multiplying  (2)  by  3, 

Adding, 

Then, 

Substituting  the  value  of  a:  in  (1), 

Then, 


=    40. 


50     45 

X  ~Y 

X       y 


74 


=    37. 


74  =  37  a;,  and  x  =  2. 

y 

-  ?  =  3,  and  «/  =  -  3. 

y 


EXAMPLES. 


Solve  the  following 


2. 


4. 


X       y 

x^  y 


10     9 


=  4. 


a;      y 
8_15^9 

X      y      2 

Sx     y      9* 
X     4:y         8 

A_±=l. 

2a;     Sy     2 

2 L^^. 

3a5     2?/     72* 


2a;  +  -  =  -ll. 
2/ 

A         3     21 

2 


2/ 


a     6_ 

6  .  a 

-  4-  -  =  c. 
a;     y 


m     n  _ 
'x~y~'^' 

a;       2/ 


hx     ay 


10. 


ax     by 
a±_b_ 

X 


1^ 

ay 


b*-a'' 
a'b'   ' 

b 

a 


ab-b'     l^a'-\-3ab 
X  y        a-\-b 


11. 


a  +  b  .  a 


=  5b-a. 


X 


«  +  *  =  2a-36. 
X     y 


ALGEBRA. 


10 


^il 


^1? 


%        J-lVV  7725 


XV.  SIMULTANEOtJS^  EQUATIONS. 


CONTAINING  MORE  THAN  TWO  UNKNOWN  ^j,  ^       3^ 
^,  QUANTITIES.  '^  ^    (^ 

170.  '^tf  we  have  three  independent  simple  equations,  con-  ^ ' 
taining  three  unknown  quantities,  we  may  combine  any  two       ^^ 
of  them  by  one  of  the  methods  of  elimination  explained  in      J^ 
§§  164  to  166,  so  as  to  obtain  a  single  equation  containing     /*^ 
only  two  unknown  quantities.  i^ 

We  may  then  combine  the  remaining  equation  with  either        r^i 
of  the  other  two,  and  obtain  another  equation  containing       (4 
the  same  two  unknown  quantities.  v'^ 

By  solving  the  two  equations  containing  two  unknown 
quantities,  we  may  obtain  their  values;  and  substituting 
them  in  either  of  the  given  equations,  the  value  of  the 
remaining  unknown  quantity  may  be  found. 

We  proceed  in  a  similar  manner  when  the  number  of 
equations  and  of  unknown  quantities  is  greater  than  three. 

The  method  of  elimination  by  addition  or  subtraction  is 
usually  the  most  convenient.    ^ 

Qx-4:y-    7z  =  17.  (1) 

dx-7y-iez  =  29.  (2) 

I0x-5y^-   3z  =  23.  (3) 


1.    Solve  the  equations 


Multiplying  (1)  by  3, 
Multiplying  (2)  by  2, 
Subtracting,  2y  +  llz=-7.  (4) 

Multiplying  (1)  by  5,        30  x  -  20 «/  -  35  s  =      85.  (5) 

Multiplying  (3)  by  3,       30a; -15y-    9z=     69.  (6) 

Subtracting  (5)  from  (6),  5y  +  26z  =  -16.  (7) 

Multiplying  (4)  by  5, 
Multiplying  (7)  by  2, 
Subtracting,  '  Sz  =  —    3. 


18x- 

-I2y 

-21z  = 

51. 

18x- 

-14?/ 

-S2z  = 

58. 

^y 

+  11;^  = 

-7. 

30x- 

-20y 

-350  = 

85. 

30x- 

-16y 

-    9z  = 

69. 

, 

52/ 

+  26z  = 

-16. 

10  2/ 

+  55;^  = 

-35. 

10  y 

+  52^  = 

-32. 

SIMULTANEOUS   EQUATIONS. 


151 


Whence, 

Substitutmg  in  (4), 
WKence, 

Substituting  in  (1), 
,^hence, 

In  certain  cases  the  solution  may  be  abridged  by  means 
of  the  artifice  which  is  employed  in  the  following  example. 


2.   Solve  the  equations 


u  -{-  X  -\-  y  =  6. 

x-\-y-\-z  =  7. 

y-{-z  -{-u=zS. 

z  -{-  u  -\-  X  =  9. 
Adding,  Su  +  Sx  +  Sy  -^  3z  =  30. 

Dividing  by  3,  u  +  x-\-y-^z  =  10. 

Subtracting  (2)  from  (6),  u  =  3. 

Subtracting  (3)  from  (5),  x  =  2. 

Subtracting  (4)  from  (6),  y  =  1. 

Subtracting  (1)  from  (5),  «  =  4. 


(1) 
(2) 
(3) 
(4) 

(5) 


Solve  the  following : 

3x-\-2y  =  lS, 
3.    |32/-2z  =  8. 
2x-Sz  =  9. 


4. 


EXAMPLES. 


6. 


3a;  4-41/4-52  = -21. 
x-\-y  —  z  =  —  ll.  7. 

y-Sz  =  -20. 

12x-^y-{-z  =  3. 

X  —  y  —  2z  =  ~l.  8. 

5x-2y  =  0. 


l2x  —  y-\-z  =  —  9. 
x  —  2y-^z  =  0. 
x-y-\-2z^-ll. 

X  —  y  -\-z  =  9. 
x-2y-\-3z  =  32. 
x  —  4:y-{-5z  =  62. 

3x  —  y  —  z  =  7. 
x-3y-z  =  21. 
x-y-3z  =  27. 


152 


ALGEBRA. 


10. 


11. 


12. 


13. 


14. 


16. 


16. 


17. 


2a;-32/  =  4. 

4:y-\-2z  =  -3. 

x-\-  2y  —  Sz  =  5. 
3x-22y-\-6z  =  4:. 
7x-    6y-Sz  =  15. 

5x  -\-    y  -\-  4:Z  =  —  5. 
3x-5y-h6z  =  -20. 
x-3y-4:Z  =  -21. 

2x-3y-4.z  =  -10. 

3x-\-4:y  +  2z  =  —  5. 
4.x-\-2y  +  3z  =  -21. 

6x-\-^y-\-3z  =  l. 
9a;-    2/ +  62  =  -39. 
^x-ly-12z  =  -2. 

2x  —  Qty  —  6z  =  —ll. 
10x  +  ^y-3z  =  m. 


18. 


19. 


20. 


21. 


x-{-3y-lz  =  31. 
3x-{-    ?/  +  5  ;2  =  —  49.   22. 
20x-\-2y-6z  =  -3b. 

9a;  +  42/  =  102-f  11. 
121/ -5^  =  6a; -9. 
15;3  4-3a;  =  -8y-16. 


5a; +  16?/ +  6:3  =  4. 

10x-\.4.y-12z  =  -l. 

[15a; -12^- 32  =  -10. 


23. 


?+   1  =  5. 
y       X 

§-+  5=1. 

X       z 


1       3^^9 

X     2y     5 

1  +  A=5 
y     3z      3 

i  +  A=I. 

z      4a;     4 


X  —  ay  =  a(a^  +  1). 


az 


—  nVn'i 


a\a'  - 1). 


y 

z  —  ax  =  —.  a^(a  +  1). 

a(x  —  c)  +  b(y  —  c)  =  0. 
6(2/  —  a)  +  c(z  —  a)  =  0. 
0(2;  —  6)  +  a(a;  —  b)=0. 

w  +  a;  +  2/  =  7. 
a;  +  2/  +  2;=-8. 
2/  +  2J  +  w  =  5. 
2;  +  tfc  +  a;  =  —  10. 

1_1      1      . 
X     y 
1_1 

y    ^ 
i_i 

2;      X 


X 

y 

2; 

1_ 

2/ 

1 

2; 

-1=6 
a; 

1 

_i_ 

-l  =  c. 

SIMULTANEOUS  EQUATIONS. 


153 


24. 


25. 


26. 


27. 


28. 


29. 


u-2x  =  -13. 
x  —  3y  =  13. 

1     1_1 

X     y~c' 

y  —  4:Z  =5. 
z  -5u  =  23. 

7x-{-4:y-3ii=0. 

30. 

y     z      a 
z^x     b 

5x-\-4:y-{-4:Z  —  5u= 

0. 

2x-{-z-u  =  0. 
2x-\-Ay—3z—u=- 

-8. 

a     c 

2     3      3 
^23 

31. 

?  +  2  =  2a. 
6      c 

|  +  ^  =  2c. 

16     a 

2     3          3 

9a; -26?/ -16:2  = - 
12a;-8i/  +  15z  =  - 

-44. 
-15. 

32. 

Z        X 

8a;-92/  +  132  =  - 

-44. 

12/     2; 

-M=^^- 

..|.6.^-±^. 

33. 

x+y+az=a-\-2. 
ay-\-az-{-a^x=a^-\-a-\-l 

[3     ^          2         2 

^az-\-ax-\-a^y=2a^-^l. 

2^3     4 

(3x-y     5y-\-4.z     19 

5       '         2       ~  2 

3     4     2 

34. 

2x-^3z     x-4:y     7 
6                4          4 

4     2     3 

4aj-2     3y-5z     49 
L      3               2            3 

154  ALGEBRA. 

XVI.  PROBLEMS. 

INVOLVING  SIMULTANEOUS  EQUATIONS. 

171.  In  solving  problems  where  two  or  more  letters  are 
used  to  represent  unknown  quantities,  we  must  obtain  from 
the  conditions  of  the  problem  as  many  independent  equations 
(§  160)  as  there  are  unknown  quantities  to  he  determined. 

172.  1.  Divide  81  into  two  parts  such  that  three-fifths 
of  the  greater  shall  exceed  live-ninths  of  the  less  by  7. 

Let  X  =  the  greater  part, 

and  y  =  the  less. 

Since  the  sum  of  the  greater  and  less  parts  is  81,  we  have 

x  +  y  =  8l.  (1) 

And  since  three-fifths  of  the  greater  exceeds  five-ninths  of  the  less 
by  7, 

f  =  f+7.  (2) 

Solving  (1)  and  (2),  x  =  45,  y  =  S6. 

2.  If  3  be  added  to  both  numerator  and  denominator  of 
a  fraction,  its  value  is  | ;  and  if  2  be  subtracted  from  both 
numerator  and  denominator,  its  value  is  ^.  Kequired  the 
fraction. 

Let  X  =  the  numerator, 

and  y  =  the  denominator. 


By  the  conditions. 


and 


X  +  3  ^  2 

y  +  s    3' 

x-2      1 


y-2      2 

Solving  these  equations,       x  =  7,  y  ==  12. 
Therefore,  the  fraction  is  ■^^. 


PROBLEMS.  155 


K^'^'^T^.^"''^ 


V 


PROBLEMS. 

'^.   Divide  59  into  two  parts  such  that  two-thirds  of  the 
^  '/I  fess  shall  be  less  by  4  than  four-sevenths  of  the  greater. 

[J\  4.   Find  two  numbers  such  that  two-fifths  of  the  greater 

v£>^ exceeds  one-half  of  the  less  by  2,  and  four-thirds  of  the  less 
us^^'  exceeds  three-fourths  of  the  greater  by  1. 

5.  If  5  be  added  to  the  numerator  of  a  certain  fraction, 
tit     its  value  is  | ;  and  if  5  be  subtracted  from  its  denominator, 

its  value  is  f.     Find  the  fraction. 

6.  If  9  be  added  to  both  terms  of  a  fraction,  its  value  is 
f ;  and  if  7  be  subtracted  from  both  terms,  its  value  is  f . 
Find  the  fraction. 

7.  A  grocer  can  sell  for  $  57  either  9  barrels  of  apples 
and  16  barrels  of  flour,  or  15  barrels  of  apples  and  14  bar- 
rels of  flour.  Find  the  price  per  barrel  of  the  apples  and 
of  the  flour. 

8.  A's  age  is  f  of  B's ;  but  in  16  years  his  age  will  be 
^  of  B's.     Find  their  ages  at  present. 

9.  If  twice  the  greater  of  two  numbers  be  divided  by 
the  less,  the  quotient  is  3  and  the  remainder  7 ;  and  if  five 
times  the  less  be  divided  by  the  greater,  the  quotient  is  2 
and  the  remainder  23.     Find  the  numbers. 

10.  If  the  numerator  of  a  fraction  be  trebled,  and  the 
denominator  increased  by  8,  the  value  of  the  fraction  is  f ; 
and  if  the  denominator  be  halved,  and  the  numerator  de- 
creased by  7,  the  value  of  the  fraction  is  \.  Find  the 
fraction. 

11.  Three  years  ago  A's  age  was  f  of  B's ;  but  in  nine 
years  his  age  will  be  ^  of  B's.     Find  their  ages  at  present. 

12.  A  and  B  can  do  a  piece  of  work  in  9  hours.  After 
working  together  7  hours,  B  finishes  the  work  in  5  hours. 
In  how  many  hours  could  each  alone  do  the  work  ? 


156  ALGEBRA. 

13.  A  man  invests  a  certain  sum  of  money  in  4^  per  cent 
stock,  and  a  sum  $  180  greater  than  the  first  in  3^  per  cent 
stock.  The  incomes  from  the  two  investments  are  equal. 
Find  the  sums  invested. 

14.  My  income  and  assessed  taxes  together  amount  to 
$  64.  If  the  income  tax  were  increased  one-fourth,  and  the 
assessed  tax  decreased  one-iifth,  they  would  together  amount 
to  $  63.80.     Find  the  amount  of  each  tax. 

16.  If  B  gives  A  $  12,  A  will  have  f  as  much  money  as 
B ;  but  if  A  gives  B  $  12,  B  will  have  J  as  much  money  as 
A.     How  much  money  has  each  ? 

16.  A  man  pays  with  a  f  5  note  two  bills,  one  of  which 
is  six-sevenths  of  the  other,  and  receives  back  in  change 
seven  times  the  difference  of  the  bills.    Find  their  amounts. 

17.  Find  three  numbers  such  that  the  first  with  one-third 
the  others,  the  second  with  one-fourth  the  others,  and  the 
third  with  one-fifth  the  others  may  each  be  equal  to  25. 

18.  A  sum  of  money  was  divided  equally  between  a  cer- 
tain number  of  persons.  Had  there  been  3  more,  each 
would  have  received  $  1  less ;  had  there  been  6  less,  each 
would  have  received  $5  more.  How  many  persons  were 
there,  and  how  much  did  each  receive  ? 

Let  X  =  the  number  of  persons, 

and  y  =  the  number  of  dollars  received  by  each. 

Then,  xy  =  the  number  of  dollars  divided. 

The  sum  of  money  could  be  divided  between  cc  +  3  persons,  each  of 
whom  would  receive  y  —  1  dollars  ;  and  between  x  —  6  persons,  each 
of  whom  would  receive  ?/  +  5  dollars. 

Whence,  (x  -I-  3)(y  -  1)  and  (x  -Q)(y  +  5)  will  also  represent  the 
number  of  dollars  divided. 

Then  (x  +  S)(y  -  1)  =  xy, 

and  (x  —  6)  (?/  -h  5)  =  xy. 

Solving  these  equations, 

X  =  12,  y  =  5. 


PROBLEMS.  157 

19.  A  man  bought  a  certain  number  of  eggs.  If  he  had 
bought  56  more  for  the  same  money,  they  would  have  cost 
a  cent  apiece  less ;  if  24  less,  a  cent  apiece  more.  How- 
many  eggs  did  he  buy,  and  at  what  price  each  ? 

20.  A  boy  spent  his  money  for  oranges.  If  he  had  got 
15  more  for  his  money,  they  would  have  cost  li  cents  each 
less;  if  5  less,  they  would  have  cost  1|-  cents  each  more. 
How  much  did  he  spend,  and  how  many  oranges  did  he  get  ? 

21.  A  sum  of  money  is  divided  equally  between  a  certain 
number  of  persons.  Had  there  been  m  more,  each  would 
have  received  a  dollars  less;  if  n  less,  each  would  have 
received  b  dollars  more.  How  many  persons  were  there, 
and  how  much  did  each  receive  ? 

22.  A  purse  contained  $6.55  in  quarter-dollars  and 
dimes;  after  6  quarters  and  8  dimes  had  been  taken  out, 
there  remained  3  times  as  many  quarters  as  dimes.  How 
many  were  there  of  each  at  first? 

23.  A  dealer  has  two  kinds  of  wine,  worth  50  and  90 
cents  a  gallon,  respectively.  How  many  gallons  of  each 
must  be  taken  to  make  a  mixture  of  70  gallons,  worth  75 
cents  a  gallons  ? 

24.  A  grocer  bought  a  certain  number  of  eggs  at  the  rate 
of  22  cents  a  dozen,  and  seven-fifths  as  many  at  the  rate  of 
14  cents  a  dozen.  He  sold  them  at  the  rate  of  20  cents  a 
dozen,  and  gained  24  cents  by  the  transaction.  How  many 
of  each  kind  did  he  buy  ? 

25.  A  and  B  can  do  a  piece  of  work  in  10  days,  A  and  C 
in  12  days,  and  B  and  C  in  20  days.  In  how  many  days 
can  each  of  them  alone  do  it  ? 

26.  A  resolution  was  adopted  by  a  majority  of  10  votes ; 
but  if  one-fourth  of  those  who  voted  for  it  had  voted  against 
it,  it  would  have  been  defeated  by  a  majority  of  6  votes. 
How  many  voted  for,  and  how  many  against  it  ? 


158  ALGEBRA. 

27.  The  sum  of  the  three  digits  of  a  number  is  13.  If 
the  number,  decreased  by  8,  be  divided  by  the  sum  of  its 
second  and  third  digits,  the  quotient  is  25;  and  if  99  be 
added  to  the  number,  the  digits  will  be  inverted.  Find  the 
number. 

Let  X  =  the  first  digit, 

y  =  the  second, 
and  z  =  the  third. 

Then,  100  x -\- 10  y  -\-  z  =  the  number, 

and  100  z  -^r  lOy  -\-x  =  the  number  with  its  digits  inverted. 

By  the  conditions  of  the  problem, 
x  +  y  +  z  =  lS, 
100x  +  lOy +  ^  -8_o^ 
y  -\-  z 
and  100x  +  10y-\-z-h99  =  100z-^10y-^x. 

Solving  these  equations,     x  =  2,  y  =  S,  z  =  3. 
Therefore,  the  number  is  283. 

28.  The  sum  of  the  two  digits  of  a  number  is  16 ;  and  if 
18  be  subtracted  from  the  number,  the  digits  will  be  inverted. 
Find  the  number. 

29.  The  sum  of  the  three  digits  of  a  number  is  23 ;  and 
the  digit  in  the  ten's  place  exceeds  that  in  the  unit's  place 
by  3.  If  198  be  subtracted  from  the  number,  the  digits  will 
be  inverted.     Find  the  number. 

30.  If  the  digits  of  a  number  of  two  figures  be  inverted, 
the  sum  of  the  resulting  number  and  twice  the  given  num- 
ber is  204 ;  and  if  the  number  be  divided  by  the  sum  of  its 
digits,  the  quotient  is  7  and  the  remainder  6.  Find  the 
number. 

31.  If  a  certain  number  be  divided  by  the  sum  of  its  two 
digits,  the  quotient  is  4  and  the  remainder  3.  If  the  digits 
be  inverted,  the  quotient  of  the  resulting  number  increased 
by  23,  divided  by  the  given  number,  is  2.    Find  the  number. 


PROBLEMS.  159 

32.  Two  vessels  contain  mixtures  of  wine  and  water.  In 
one  there  is  three  times  as  much  wine  as  water,  and  in  the 
other  five  times  as  much  water  as  wine.  How  many  gallons 
must  be  taken  from  each  to  fill  a  third  vessel  whose  capacity 
is  7  gallons,  so  that  its  contents  may  be  half  wine  and  half 
water  ? 

33.  If  a  lot  of  land  were  6  feet  longer  and  5  feet  wider, 
it  would  contain  839  square  feet  more ;  and  if  it  were  4  feet 
longer  and  7  feet  wider,  it  would  contain  879  square  feet 
more.     Find  its  length  and  width. 

34.  A  and  B  are  building  a  piece  of  fence  189  feet  long. 
After  9  hours  A  leaves  off,  and  B  finishes  the  work  in  12 1 
hours.  If  12  hours  had  occurred  before  A  left  off,  B  would 
have  finished  the  work  in  4^  hours.  How  many  feet  does 
each  build  in  one  hour  ? 

35.  The  sum  of  the  three  digits  of  a  number  is  17.  The 
sum  of  3  times  the  first  digit,  5  times  the  second,  and  4 
times  the  third  is  70 ;  and  if  297  be  added  to  the  number, 
the  digits  will  be  inverted.     Find  the  number. 

36.  The  rate  of  an  express  train  is  five-thirds  that  of 
a  slow  train,  and  it  travels  36  miles  in  32  minutes  less  time 
than  the  slow  train.    Find  the  rate  of  each  in  miles  an  hour. 

37.  Divide  $396  between  A,  B,  C,  and  D  so  that  A  may 
receive  one-half  the  sum  of  the  shares  of  B  and  C,  B  one- 
third  the  sum  of  the  shares  of  C  and  D,  and  C  one-fourth 
the  sum  of  the  shares  of  A  and  D. 

38.  A  merchant  has  two  casks  of  wine,  containing  together 
56  gallons.  He  pours  from  the  first  into  the  second  as  much 
as  the  second  contained  at  first;  he  then  pours  from  the 
second  into  the  first  as  much  as  was  left  in  the  first ;  and 
again  from  the  first  into  the  second  as  much  as  was  left  in 
the  second.  There  is  now  three-fourths  as  much  in  the  first 
as  in  the  second.  How  many  gallons  did  each  contain  at 
first? 


160  ALGEBRA. 

39.  A  crew  can  row  10  miles  in  50  minutes  down  stream, 
and  12  miles  in  an  hour  and  a  half  against  the  stream. 
Find  the  rate  in  miles  an  hour  of  the  current,  and  of  the 
crew  in  still  water. 

Let  X  —  the  number  of  miles  an  hour  rowed  by  the  crew  in 

still  water, 
and  y  —  the  number  of  miles  an  hour  of  the  current. 

Then,  x  ■\ij  —  the  number  of  miles  an  hour  of  the  crew  rowing 

down  stream, 

and  X  —  y  =  the  number  of  miles  an  hour  of  the  crew  rowing  up 

stream. 
Since  the  number  of  miles  an  hour  rowed  by  the  crew  is  equal  to 
the  distance  divided  by  the  time  in  hours,  we  have 

D 

and  x-w=12-^-  =  8. 

^  2 

Solving  these  equations,         a:  =  10,  y  =  2. 

40.  A  crew  can  row  a  miles  in  m  hours  down  stream,  and 
b  miles  in  ii  hours  against  the  stream.  Find  the  rate  in 
miles  an  hour  of  the  current,  and  of  the  crew  in  still  water. 

41.  A  vessel  can  go  63  miles  down  stream  and  back  again 
in  20  hours ;  and  it  can  go  3  miles  against  the  current  in 
the  same  time  that  it  goes  7  miles  with  it.  Find  its  rate  in 
miles  an  hour  in  going,  and  in  returning. 

42.  If  a  number  of  two  figures,  diminished  by  3,  be 
divided  by  the  sum  of  its  digits,  the  quotient  is  5.  If  the 
digits  be  inverted,  the  quotient  of  the  resulting  number 
increased  by  18,  divided  by  the  sum  of  the  digits,  is  7. 
Find  the  number. 

43.  The  digits  of  a  number  of  three  figures  have  equal 
differences  in  their  order.  If  the  number  be  divided  by 
one-half  the  sum  of  its  digits,  the  quotient  is  41 ;  and  if 
594  be  added  to  the  number,  the  digits  will  be  inverted. 
Find  the  number. 


PROBLEMS.  161 

44.  If  I  were  to  make  my  field  5  feet  longer  and  7  feet 
wider,  its  area  would  be  increased  by  830  square  feet ;  but 
if  I  were  to  make  its  length  8  feet  less,  and  its  width  4  feet 
less,  its  area  would  be  diminished  by  700  square  feet.  Find 
its  length  and  width. 

45.  A  certain  sum  of  money  at  simple  interest  amounts 
in  3  years  to  $  420,  and  in  7  years  to  $  480.  Required  the 
sum  and  the  rate  of  interest. 

46.  A  certain  sum  of  money  at  simple  interest  amounts 
in  m  years  to  a  dollars,  and  in  n  years  to  b  dollars.  Re- 
quired the  sum  and  the  rate  of  interest. 

47.  A  and  B  together  can  do  a  piece  of  work  in  8 J  days ; 
but  if  A  had  worked  f  as  fast,  and  B  |  as  fast,  they  would 
have  done  it  in  7|  days.  In  how  many  days  could  each 
alone  do  the  work  ? 

48.  A  sum  of  money  at  simple  interest  amounts  to  $  2080 
in  8  months,  and  to  $  2150  in  15  months.  Find  the  sum 
and  the  rate  of  interest. 

49.  A  train  running  from  A  to  B  meets  with  an  accident 
which  causes  its  speed  to  be  reduced  to  one-third  of  what 
it  was  before,  and  it  is  in  consequence  5  hours  late.  If  the 
accident  had  happened  60  miles  nearer  B,  the  train  would 
have  been  only  1  hour  late.  Find  the  rate  of  the  train 
before  the  accident,  and  the  distance  to  B  from  the  point 
of  detention. 

Let    3  a;  =  the  number  of  miles  an  hour  of  the  train  before  the 

accident. 
Then,  x  =  the  number  of  miles  an  hour  after  the  accident. 
Let       y  =  the  number  of  miles  to  B  from  the  point  of  detention. 

The  train  would  have  done  the  last  y  miles  of  its  journey  in  ^ 

V  ^^ 

hours  ;  but  owing  to  the  accident,  it  does  the  distance  in  -  hours. 

T"-'  l  =  ii  +  ^-  CD 


162  ALGEBRA. 

If  the  accident  had  occurred  60  miles  nearer  B,  the  distance  to  B 
from  the  point  of  detention  would  have  been  y  —  QO  miles. 

Had  there  been  no  accident,  the  train  would  have  done  this  in 

^ —  hours,  and  the  accident  would  have  increased  the  time  to 

3x  X 

hours. 

Then,  — =  ^ +  1.  (2) 

X  Sx  ^  ^ 

Subtracting  (2)  from  (1),   —  =  ^—+4,  or  —  =  4. 

X        o  X  X 

Whence,  x  =  10. 

Then  the  rate  of  the  train  before  the  accident  was  30  miles  an  hour. 

Substituting  in  (1),  ^  =  ^  +  ^'  ^^  T^  =  ^• 

Whence,  y  z=76. 

50.  A  train  running  from  A  to  B  meets  with  an  accident 
which  delays  it  45  minutes ;  it  then  proceeds  at  five-sixths 
its  former  rate,  and  arrives  at  B  75  minutes  late.  Had  the 
accident  occurred  45  miles  nearer  A,  the  train  would  have 
been  90  minutes  late.  Find  the  rate  of  the  train  before  the 
accident,  and  the  distance  to  B  from  the  point  of  detention. 

51.  The  unit's  digit  of  a  number  of  three  digits  is  7.  If 
the  digits  in  the  hundreds'  and  tens'  places  be  interchanged, 
the  number  is  decreased  by  180.  If  the  digit  in  the  hun- 
dreds' place  be  halved,  and  the  other  two  digits  interchanged, 
the  number  is  decreased  by  273.     Find  the  number. 

52.  A,  B,  C,  and  D  play  at  cards,  having  together  $  46. 
After  A  has  won  one-third  of  B's  money,  B  one-fourth  of 
C's,  and  C  one-fifth  of  D's,  A,  B,  and  C  have  each  $  10. 
How  much  had  each  at  first? 

53.  A,  B,  and  C  have  together  f  24.  A  gives  to  B  and 
C  as  much  as  each  of  them  has;  B  gives  to  A  and  C  as 
much  as  each  of  them  then  has ;  and  C  gives  to  A  and  B 
as  much  as  each  of  them  then  has.  They  have  now  equal 
amounts.     How  much  did  each  have  at  first  ? 


PROBLEMS.  163 

54.  The  fore-wheel  of  a  carriage  makes  8  revolutions 
more  than  the  hind-wheel  in  going  180  feet ;  but  if  the  cir- 
cumference of  the  fore- wheel  were  |  as  great,  and  of  the 
hind- wheel  f  as  great,  the  fore- wheel  would  make  only  5 
revolutions  more  than  the  hind-wheel  in  going  the  same 
distance.     Find  the  circumference  of  each  wheel. 

55.  A  and  B  together  can  do  a  piece  of  work  in  m  days, 
B  and  C  in  n  days,  and  C  and  A  in  p  days.  In  what  time 
can  each  alone  perform  the  work  ? 

56.  A  piece  of  work  can  be  completed  by  A  working  3 
days,  B  7  days,  and  C  1  day ;  by  A  working  5  days,  B  1 
day,  and  C  7  days;  or  by  A  working  1  day,  B  5  days,  and 
C  11  days.  In  how  many  days  can  each  alone  perform  the 
work  ? 

57.  A  man  has  a  sum  of  money  invested  at  a  certain  rate 
of  interest.  Another  man  has  a  sum  greater  by  $  3000, 
invested  at  a  rate  1  per  cent  less,  and  his  income  is  $  45 
less  than  that  of  the  first.  A  third  man  has  a  sum  less  by 
$  2000  than  that  of  the  first,  invested  at  a  rate  1  per  cent 
greater,  and  his  income  is  f  40  greater  than  that  of  the  first. 
Find  the  capital  of  each  man,  and  the  rate  at  which  it  is 
invested. 

58.  A  and  B  can  do  a  piece  of  work  in  a  hours.  After 
working  together  b  hours,  B  finishes  the  work  in  c  hours. 
In  how  many  hours  could  each  alone  do  the  work  ? 

59.  A  crew  row  up  stream  26  miles  and  down  stream  35 
miles  in  9  hours.  They  then  row  up  stream  32  miles  and 
down  stream  28  miles  in  10  hours.  Find  the  rate  in  miles 
an  hour  of  the  current,  and  of  the  crew  in  still  water. 

(Let  X  and  y  represent  the  number  of  miles  an  hour  of  the  crew 
rowing  up  and  down  stream,  respectively.) 

60.  A  sum  of  money,  at  6  per  cent  interest,  amounts  to 
$  5900  for  a  certain  time,  and  to  ^  7100  for  a  time  longer 
by  4  years.     Find  the  principal  and  the  time. 


164  ALGEBRA. 

61.  A  gives  to  B  and  C  twice  as  much  money  as  each  of 
them  has;  B  gives  to  A  and  C  twice  as  much  as  each  of 
them  then  has  ;  and  C  gives  to  A  and  B  twice  as  much  as 
each  of  them  then  has.  Each  has  now  f  27.  How  much 
did  each  have  at  first  ? 

62.  A  party  at  a  tavern  found,  on  paying  their  bill,  that 
had  there  been  4  more,  each  would  have  paid  75  cents  less ; 
but  if  there  had  been  4  less,  each  would  have  paid  $  1.50 
more.    How  many  were  there,  and  how  much  did  each  pay  ? 

63.  An  express  train  travels  30  miles  in  27  minutes  less 
time  than  a  slow  train.  If  the  rate  of  the  express  train 
were  f  as  great,  and  of  the  slow  trairi  f  as  great,  the  express 
train  would  travel  30  miles  in  54  minutes  less  time  than 
the  slow  train.     Find  the  rate  of  each  in  miles  an  hour. 

64.  A  and  B  run  a  race  of  450  feet.  The  first  heat,  A 
gives  B  a  start  of  135  feet,  and  is  beaten  by  4  seconds ;  the 
second  heat,  A  gives  B  a  start  of  30  feet,  and  beats  him  by 
3  seconds.     How  many  feet  can  each  run  in  a  second  ? 

65.  A  sum  of  money  consists  of  quarter-dollars,  dimes, 
and  half-dimes.  Its  value  is  as  many  dimes  as  there  are 
pieces  of  money  ;  its  value  is  also  as  many  quarters  as  there 
are  dimes ;  and  the  number  of  half-dimes  is  one  more  than 
the  number  of  dimes.     Find  the  number  of  each  coin. 

66.  A  man  invests  $  5100,  partly  in  3|-  per  cent  stock  at 
$  90  a  share,  and  partly  in  4  j3er  cent  stock  at  $  120  a  share, 
the  par  value  of  each  share  being  ^100.  If  his  annual 
income  is  ^185,  how  many  shares  of  each  stock  does  he 
buy? 

67.  A  and  B  run  a  race  of  336  yards.  The  first  heat,  A 
gives  B  a  start  of  28  yards,  and  beats  him  by  2  seconds ; 
the  second  heat,  A  gives  B  a  start  of  12  seconds,  and  is 
beaten  by  48  yards.  How  many  yards  can  each  run  in  a 
second  ? 


INEQUALITIES.  165 


XVII.    INEQUALITIES. 

173.  Definitions. 

The  Signs  of  Inequality,  >  and  <,  are  read  "is  greater 
than  "  and  "  is  less  than,"  respectively. 

Thus,  a  >  6  is  read  "  a  is  greater  than  b" ;  a  <  6  is  read 
"  a  is  less  than  5." 

The  Sign  of  Continuation,  ••♦,  signifies  "and  so  on/^  or 
"  continued  by  the  same  law."  * 

174.  One  number  is  said  to  be  greater  than  another  when 
the  remainder  obtained  by  subtracting  the  second  from  the 
first  is  a  positive  number ;  and  one  number  is  said  to  be  less 
than  another  when  the  remainder  obtained  by  subtracting 
the  second  from  the  first  is  a  negative  number. 

Thus,  if  a  —  6  is  a  positive  number,  a  >  6 ;  and  if  a  —  6 
is  a  negative  number,  a  <b. 

175.  An  Inequality  is  a  statement  that  one  of  two  expres- 
sions is  greater  or  less  than  another. 

The  First  Member  of  an  inequality  is  the  expression  to 
the  left  of  the  sign  of  inequality ;  the  Second  Member  is  the 
expression  to  the  right  of  that  sign. 

Any  term  of  either  member  of  an  inequality  is  called  a 
term  of  the  inequality. 

176.  Two  or  more  inequalitiea^are  said  to  s^ibsist  in  the 
same  sense  when  the  first  member  is  the  greater  or  the  less 
in  both. 

Thus,  a >  6  and  c>d  subsist  in  the  same  sense. 

177.  An  inequality  will  continue  in  the  same  sense  after 
the  same  quantity  has  been  added  to,  or  subtracted  from,  both 
members. 


166  ALGEBRA. 

For  consider  the  inequality  a^b. 

Then  by  §  174,  a  —  b  is  a  positive  number. 

Hence,  each  of  the  numbers 

(a  +  c)  —  (6  +  c),  and  (a  —  c)  —  (6  —  c) 
is  positive,  since  each  is  equal  to  a  —  b. 

Therefore,    a  -f  c  >  6  -f-  c,  and  a  —  c>  b  —  c.         (§  174) 

178.   It  follows  from  §  177  that  a  term  may  be  transposed 
from  one  member  of  an  inequality  to  the  other  by  changing  its 


179.  If  the  signs  of  all  the  terms  of  an  inequality  be  changed, 
the  sigyi  of  inequality  must  be  reversed. 

For  consider  the  inequality  a  —  b  >  c  —  d. 
Transposing  every  term,  we  have 

d-c>b-a.  (§  178) 

That  is,  b  —  a<d  —  c. 

180.  An  inequality  will  continue  in  the  same  sense  after 
both  members  have  been  multiplied  or  divided  by  the  same 
positive  number. 

For  consider  the  inequality  a>b. 

By  §  174,  a  —  b  is  a  positive  number. 

Hence,  if  m  is  a  positive  number,  each  of  the  numbers 

m(a  —  b)  and , 

or,  ma  —  mb  and ,  is  positive. 

'  mm 

Therefore,  ma  >  mb,  and  —  >  — 

'  'mm' 

181.  It  follows  from  §§  179  and  180  that  if  both  members 
of  ayi  inequality  be  multiplied  or  divided  by  the  same  negative 
number,  the  sign  of  inequality  must  be  reversed. 


INEQUALITIES.  167 

182.  If  any  number  of  inequalities,  subsisting  in  the  same 
sense,  be  added  meynber  to  member,  the  resulting  inequality 
will  also  subsist  in  the  same  sense. 

For  consider  the  inequalities  a>b,  a'  >  b',  a"  >  b",  •••. 
Then  each  of  the  numbers  a  —  b,  a'  —  b',  a"  —  b",  •••,  is 
positive. 

Therefore,  their  sum 

a-b  +  a'-b'-\-a"-b"+'", 

or,  a  -f-  a'  +  a"  H (6  +  6'  +  b"  +  .••), 

is  a  positive  number. 

Whence,     a-^a'  -\-  a"  -+-...>  6  -f-  6'  +  b"  +  •••. 

183.  It  is  to  be  observed  that,  if  two  inequalities,  subsist- 
ing in  the  same  sense,  be  subtracted  member  from  member, 
the  resulting  inequality  does  not  necessarily  subsist  in  the 
same  sense. 

Thus,  if  a>6  and  a'>b',  the  numbers  a—b  and  a'—b' 
are  positive. 

But  (a  —  b)  —  (a'  —  b'),  or  its  equal  (a— a')—  (b  —  b'),  may 
be  positive,  negative,  or  zero ;  and  hence  a— a'  may  be  greater 
than,  less  than,  or  equal  to  b  —  b'. 

EXAMPLES. 

184.  1.   Find  the  limit  of  x  in  the  inequality 

'-!<¥-»■ 

Multiplying  both  members  by  3  (§  180),  we  have 

21x-23<2x  +  16. 
Transposing  (§  178),  and  uniting  terms, 

19a;  <  38. 
Dividing  both  members  by  19  (§  180), 

X  <  2,     Ans. 


168  ALGEBRA. 

2.   Find  the  limits  of  x  and  y  in  the  following: 

Sx-j-2y>37.  (1) 


6x> 

45. 

x>2 

1. 

Qx 

+  ^y> 

74. 

6x 

+  9y  = 

99. 

2x-{-3y  =  33.  (2) 

Multiplying  (1)  by  3,  .  9x  +  6ij>  111. 

Multiplying  (2)  by  2,  4  x  +  6 «/  =    66. 

Subtracting  (§  177), 
Whence, 

Multiplying  (1)  by  2, 
Multiplying  (2)  by  3, 

Subtracting,  —  5  ?/  >  —  25. 

Dividing  both  members  by  —  5  (§  181),  y  <  5.  -^ 

Find  the  limits  of  x  in  the  following : 

3.  (6x-iy-2S<(4.x-3)i9x-\-2). 

4.  (3a;  +  2)(4a;-5)>(2a;-3)(6a;-hl)  +  5. 

5.  (5x  +  iy  +  W>(3x-2y-{-(4.x-{-3f. 

6.  (x  -2){x-  3)  (a;  -h  4)  <  (a;  + 1)  (a;  +  2)  (a;  -  4). 

7.  6  ma;  —  5  an  >  15  am  —  2  Tiic,  if  3  m  +  n  is  negative. 

8.  — i ^^  <  2,  if  a  and  b  are  positive,  and  a>  b. 

a  b 

Find  the  limits  of  x  and  ?/  in  the  following : 

9_    i4.x  +  9y<4:0.  ^^    (7x-^2y>25. 

\6x-y  =  2.  '   \3x-\-5y  =  19. 

11.  Find  the  limits  of  x  when 

5a;  +  7<9a;-13,  and  11a;  -  20  <  6a;  +  25. 

12.  A  certain  positive  integer,  plus  21,  is  greater  than  8 
times  the  number,  minus  35 ;  and  twice  the  number,  plus 
11,  is  less  than  7  times  the  number,  minus  19.  Find  the 
number. 


INEQUALITIES.  169 

13.  A  teacher  has  a  number  of  his  pupils  such  that  8  times 
their  number,  minus  31,  is  less  than  3  times  their  number, 
plus  69 ;  and  13  times  their  number,  minus  45,  is  greater  than 
7  times  their  number,  phis  57.     How  many  pupils  has  he  ? 

14.  A  shepherd  has  a  number  of  sheep  such  that  4  times 
the  number,  minus  7,  is  greater  than  6  times  the  number, 
minus  89 ;  ^nd  5  times  the  number,  plus  3,  is  greater  than 
twice  the  number,  plus  114.     How  many  sheep  has  he  ? 

15.  Prove  that  if  a  and  b  are  unequal  positive  numbers, 

K  +  ->2- 
0      a 

Since  the  square  of  any  number  is  positive, 

(a-6)2>0. 

That  is,  a2-2a6  +  &2>0. 

Transposing  -  2  ab,  a^  ■^b'^>2  ah. 

Dividing  each  term  of  the  inequality  by  a6  (§  180) ,  we  have 

^+^>2. 
h      a 

16.  Prove  that  for  any  value  of  x,  except  1,  x^  +  1  >  2  x. 

17.  Prove  that  for  any  value  of  x,  except  |,  9  a;^  -f-  4  >  12  a;. 

"  In  each  of  the  fallowing  examples,  the  letters  are  under- 
stood as  representing  positive  numbers. 

18.  Prove  that  -^  +  ?^  >  2,  if  6  is  not  equal  to  \  a. 

26       a 

19.  Prove  that  (ci  +  26)(a  -  26)  >  6(6a  -  13  6),  if  h  is 
not  equal  to  \a. 

20.  Prov.3  that  a(9a  -  46)  >46(2a  -  6),  if  h  is  not 
equal  to  fa.  / 

21.  Prove  that  (a^  -  W)  {c"  -  dF)  <  (ac  -  hdf,  if  he  does 
not  equal  ad. 


170  /-  ALGEBRA. 


XVIII.  INVOLUTION. 

185.  Involution  is  the  process  of  raising  a  given  expres- 
sion to  any  required  power  whose  exponent  is  a  positive 
integer. 

This  may  be  effected,  as  is  evident  from  §  6,  by  taking 
the  expression  as  a  factor  as  many  times  as  there  are  units 
in  the  exponent  of  the  required  power. 

INVOLUTION  OF  MONOMIALS. 

186.  1.   ¥mdtheY3i\iieoi(5a%y. 

By  §  6,  (5a352)3  ^6(^352  x  6a^b^  x  ^a%^  =  126  a^¥,  Ans. 

2.  Find  the  value  of  (-  ay. 

(-ay=  (-  a)  X  (-  a)  X  (-  a)  X  {-  a)=  a^  (§49),  Ans. 

3.  Find  the  value  of  (-  3  my. 

(-3m4)3=  (-3TO4)x(-3TO4)x(-3m4)  =  -27mi2(§49),  Aiis. 

From  the  above  examples,  we  derive  the  following  rule  : 

liaise  the  absolute  value  of  the  numerical  coefficient  to  the 
required  power,  and  multiply  the  exponent  of  each  letter  by  the 
expoyient  of  the  required  power. 

Give  to  every  power  of  a  positive  term,  and  to  every  Even 
poiver  of  a  negative  term,  the  positive  sign,  and  to  every  Odd 
power  of  a  negative  term  the  negative  sign. 

EXAMPLES. 
Find  the  values  of  the  following : 

4.  (a'b^c'f.  8.    (2m^n'y.  12.  (pq'^r'y\ 

5.  {:f^fzy\  9.    (-a26V)l  13.  {-QxY^y. 

6.  {-mhipy.    .      10.    {x^yz'y.  14.  {^a'^x'^. 

7.  (-12a2«63n-^2^      11.    (^-Sx'y'^y.  15.  (-5m/iy)^ 


INVOLUTION.  171 

A  fraction  may  be  raised  to  any  required  power  by  rais- 
ing both  niwierator  and  denominator  to  the  required  2)ower, 
and  dividing  the  first  result  by  the  second. 

16.   Find  the  value  oi  ( -  — Y- 
V     32/V 

V       3y2;  (3j/2)4         812/8' 

Find  the  values  of  the  following : 

"(f^"    -C-^)'    "-(^S)' 

SQUARE  OF  A  POLYNOMIAL. 
187.   We  find  by  multiplication : 

a  -\-b  -\-  c 
a  -\-b  -\-  c 


a^  +    ab  +    ac 

-\-    ab  -\-b^-\-    be 

-\-    ac         -\-    bc-\-  <? 

a^  +  2a6 -f  2ac  +  62  4- 26c  +  c2 

The  result,  for  convenience  of  enunciation,  may  be  written : 

(a  +  6  +  c)2  =  a^  +  62  +  c2  +  2  a6  +  2  ac  +  2  6c. 

In  like  manner  we  find : 

(a  +  6  +  c  +  d)2  =  a-  +  62  +  c2  +  ^2 

4-2a6  +  2ac  +  2ad  +  26c  +  26d  +  2c(i; 

and  so  on. 

We  then  have  the  following  rule : 

Tlie  square  of  a  polynomial  is  equal  to  the  sum  of  the  squares 
of  its  terms,  plus  twice  the  product  of  each  term  by  ea^h  of  the 
following  terms. 


172  ALGEBRA. 

EXAMPLES. 
1.    Square  2  a^  —  3  a;  —  5. 

The  squares  of  the  terms  are  4x*,  9x'^,  and  25. 
Twice  the  first  term  into  each  of  the  following  terms  gives  the 
results  -  12  x^  and  -  20  x^. 

Twice  the  second  term  into  the  following  term  gives  the  result  30  x. 

Then,    (2x2  -  3x  -  5)2  =  4x4  +  9x2  +  25  -  12x3  -  20x2  +  30x 
=  4x*  -  12x3  -  11  x2  +  30x  +  25,  Ans. 

Square  each  of  the  following : 


2. 

a  —  b  —  c. 

11. 

a'-4:ab-j-3b\ 

3. 

x-y-j-z. 

12. 

2  x^  -{-  3  xy  -^  y^. 

4. 

x'  +  2x-\-l. 

13. 

a^^Qx'-7. 

.5. 

m  +  2n  —  Sp. 

—  14. 

4a*-5aV-3x«. 

6. 

2a2-a  +  4. 

15. 

a  —  b  —  c  —  d. 

7. 

5a^-3x-l. 

16. 

a  -\-  b  —  c  -{-  d. 

8. 

3x'-\-4.x-i-2. 

'      17. 

g^-r^-X  +   2. 

9. 

6n^-\-n  —  5. 

.  18. 

a^  +  2  a^  -  3  a  -  4. 

10. 

2a-5b-G. 

^  19. 

2iB8_5a^  +  4a;-3. 

CUBE  OF  A  BINOMIAL. 

188.   We  find  by  multiplication  : 

(a  +  6)2=:a2-f-2a&  +  &' 
a  +  & 


-bf: 

a' 

+  2a26+    ab' 

a?b  +  2ab''  +  W 

(a 
(a 

a 

+  3  a^ft  -f  3  aW  +  b^ 

-  2  a5  +  6^ 

-b 

a^ 

-2a^b-{-    ab^ 

-    a26  +  2a62_63 

(a  -by  =  a^-3  a'b  +  3  a^^  -  b^ 


INVOLUTION.  173 

That  is,  the  cube  of  the  sum  of  two  quantities  is  equal  to 
the  cube  of  the  first,  p??ts  three  times  the  square  of  the  first 
times  the  second,  plus  three  times  the  first  times  the  square  of 
the  second,  plus  the  cube  of  the  second. 

The  cube  of  the  difference  of  two  quantities  is  equal  to 
the  cube  of  the  first,  minus  three  times  the  square  of  the  first 
times  the  second,  plus  three  times  the  first  times  the  square 
of  the  second,  minus  the  cube  of  the  second. 

EXAMPLES. 

1.  Find  the  cube  of  a  +  2  6. 

We  have,  (a  +  2  6)8  =  a^  +  3  a'^(2  b)+S  a(2  6)2  +  (2  6)3 
=  a^  +  6a^b  +  12  ab'^  +  8  63,  Ans. 

2.  Find  the  cube  of  2  ar^  -  3  /. 

(2  x3  -  3  y2)8  =  (2  x3)8  -  3(2  a;8)2(3  y^)  +  3(2  x^)  (3  y^y  -  (3  y^y 
=  8  x9  -  36  o^y^  +  54  x^y^  -  27  y^,  Ans. 

Cube  each  of  the  following : 

3.  ax-\-by.               7.   oc^  +  5.  11.  Sa.-^-Sx. 

4.  x  +  2.                 8.   6a- b.  12.  4:X*-^oyz^ 
r— 5.   3a-l.             ^9.   5x-^2y.  >-13.  2a;-7a;3. 

6.   m-4?i.  10.   4m-3n^  14.   5a«  +  66^ 

The   cube   of   a  trinomial  may  be  found  by  the  above 

method,  if  two  of  its  terms  be  enclosed  in  a  parenthesis 

and  regarded  as  a  single  term. 

15.   Find  the  cube  of  ar^  —  2  a;  —  1. 

(x2  -  2 X  -  1)3  =  [(a;2  -  2  x)  -  l.]3 

=  (x2_2x)8-3(x2-2x)2  +  3(x2-2x)-  1 
=  x6-6x5+12x4-8x3-3(x*-4x3+4x2)  +  3(x2-2x)-l 
=  x6-6x5+12x4-8a;3-3x*+12x3_i2x2+3x2-6a;-l 
=  x6-6x5  + 9x4  + 4x3-9x2-6x- 1,  Afis. 

.    Cube  each  of  the  following : 
^16.   a-{-b-c.  18.   x-^J-\-2z.  20.   2x^-\-x-3. 

17.   ay'  +  x  +  l.         ^9.   a2-3a-l.       21.   3-4a;  +  ar^. 


174  ALGEBRA. 


XIX.  EVOLUTION. 

189.  If  an  expression  when  raised  to  the  7ith  power,  w 
being  a  positive  integer,  is  equal  to  another  expression,  the 
first  expression  is  said  to  be  the  nth.  Root  of  the  second. 

Thus,  if  a""  =  b,  a  is  the  nth  root  of  b. 

190.  Evolution  is  the  process  of  finding  any  required 
root  of  an  expression. 

191.  The  Radical  Sign,  y,  when  written  before  an  ex- 
pression, indicates  some  root  of  the  expression. 

Thus,  Va  indicates  the  second^  or  square  root  of  a ; 
Va  indicates  the  third,  or  cube  root  of  a ; 
Va  indicates  the  fourth  root  of  a ;  and  so  on. 

The  index  of  a  root  is  the  number  written  over  the  radical 
sign  to  indicate  what  root  of  the  expression  is  taken. 
If  no  index  is  expressed,  the  index  2  is  understood. 

EVOLUTION  OF  MONOMIALS. 

192.  1.   Required  the  cube  root  of  a^b^c^. 

We  have,  (abH^)^  =  a%^(^. 

Whence,  \/a%^c^  =  ahH^.  (§  189) 

2.  Required  the  fifth  root  of  —  32  a^. 
We  have,  (  -  2  a)^  =  -  32  a^. 
Whence,                        \/-32a5  =  -2a. 

3.  Required  the  fourth  root  of  a^. 

We  have  either  (+  a)*  or  (—  a)*  equal  to  a^. 
Whence,  Va!^  =  ±a. 

The  sign  ±,  called  the  double  sign,  is  prefixed  to  an  ex- 
pression when  we  wish  to  indicate  that  it  is  either  -f  or  — . 


EVOLUTION.  175 

193.   From  §  192  we  derive  the  following  rule : 
Extract  the  required  root  of  the  absolute  value  of  the  numeri- 
cal coefficient,  and  divide  the  exponent  of  each  letter  by  the 
index  of  the  required  root. 

Give  to  every  even  root  of  a  positive  term  the  sign  ± ,  ayid 
to  every  odd  root  of  any  term  the  sign  of  the  term  itself. 

EXAMPLES. 
1.   Find  the  square  root  of  9(X*6V^. 


By  the  rule,  V9  a^ft^cio  =  ^  3  a%^&,  Ans. 

2.   Find  the  cube  root  of  —  64  Qifiy^". 


V—  64 a^y^'*  =  —  4 x^y,  Ans. 
Find  the  values  of  the  following : 


■3.    V49a«6^. 

4.    ^125a!«2/'. 


5.  V  —  m}^nJp^^. 

6.  -yiGA^. 


9. 
10. 

^243a^6». 

--11. 

<J-m2m''n^Y\ 

12. 

Vl44a*^a^+«. 

13. 

^-729a:3m-6^ 

—  14. 

</256a«-6'^ 

—  7.    V64a:>«y^«;2". 

To  find  any  root  of  a  fraction,  extract  the  required  root  of 
both  numerator  and  denominator,  and  divide  the  first  result 
by  the  second. 

15.   Find  the  value  of  ^'/-?I^. 

We  have,      ;C^1^  =  -<^^^  =  -^-^,  Ans. 

Find  the  values  of  the  following : 

i«       liea^  i«      4/81mV^  „n      6/64m^ 


,7^   ^^343^^  13_     JZ^  21.   VC| 

\      64  \      32  2^^  \128 


176  ALGEBRA. 

The  root  of  a  large  number  may  sometimes  be  found  by 
resolving  it  into  its  prime  factors. 

22.   Find  the  square  root  of  254016. 


We  have,    \/254016  =  V26  x  3*  x  72  =  2^  x  82  x  7  =  504,  Ans. 

23.   Find  the  value  of  ^^72  x  75  x  135. 

We  have,  ^72  x  75  x  136  =  y/(2»  x  32)  x  (3  x  52)  x  (S^  x  5) 


=  V28  x  36  X  58  =  2  X  32  X  5  =  90,  Ans. 
Find  the  values  of  the  following : 


"^4.    V3l36.       26.    V63504.  28.    V42  x  56  x  147. 


25.    V18225.      27.    V48  x  54x72.-29.    ^13824 


30.    Vl5a6x216cx35ca.  31.    V  213444. 

32.    ^91125.      33.    a/20736.  34.   a/7776. 

-^  35.    a/63  X  162  X  196.  —  36.    a/56  x  98  x  112. 

37.    V(a2  -^5a-\-  6){d'  +  2a  -  3)(a2  -\-a-2).     . 

SQUARE  ROOT  OF  A  POLYNOMIAL. 

194.  Since  (a  +  by  =  a^ -^2ab -\-  b%  we  know  that  the 
square  root  of  a^  ■}- 2  ab  -\- b'^  is  a  +  b. 

It  is  required  to  find  a  process  by  which,  when  the  e;c- 
pression  a^  -{-2ab  +  b^  is  given,  its  square  root  may  be 
determined. 

a^  -\-2ab  -^b^    a  4-  6        The  first  term  of  the  root,  a,  is  found 
^2  by  taking  the  square  root  of  the  first 

"^ — ,        ,  2  term  of  the  given  expression. 

Subtracting  the  square   of   a  from 
Z  ao  -\-  0  ^jjg  given  expression,  the  remainder  is 

2a&  +  6'^  or  (2a  +  h)b. 
If  we  divide  the  first  term  of  this  remainder  by  2  a,  that  is,  by  twice 
the  first  term  of  the  root,  we  obtain  the  second  term  of  the  root,  6. 


2a-\-b 


EVOLUTION.  177 

Adding  this  to  2  a,  we  obtain  the  complete  divisor,  2  a  +  6. 
Multiplying  this  by  6,  and  subtracting  the  product,  2  a6  -f  6'^,  from 
f he  remainder,  there  are  no  terms  remaining. 

From  the  above  process,  we  derive  the  following  rule : 

Arrange  the  expression  according  to  the  powers  of  some 
letter. 

Extract  the  square  root  of  the  first  temij  write  the  result  as 
the  first  term  of  the  root,  and  subtract  its  square  from  the  given 
expression,  arranging  the  remainder  in  the  same  ordei'  of  pow- 
ers as  the  given  expression. 

Divide  the  first  term  of  the  remainder  by  twice  the  first  term 
of  the  root,  and  add  the  quotient  to  the  part  of  the  root  already 
found,  and  also  to  the  trial-divisor. 

Multijyly  the  complete  divisor  by  the  term  of  the  root  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  terms  remain,  proceed  as  before,  doubling  the  part 
of  the  root  already  found  for  the  next  trial-divisor. 

EXAMPLES.  '^^^ ^^Ji^ 

195.   1.   Find  the  square  root  of^9  a^-  30  a^3i?  +  25  a\ 


9a;4-30a*x2  +  25a' 


6*2  _  6 a* 


-30a8x2  +  25a6 
-  30  a3x2  +  26  a6 


The  first  term  of  the  root  is  the  square  root  of  9  x*  or  3  x^. 

Subtracting  the  square  of  3  x^,  or  9  x*,  from  the  given  expression, 
the  first  term  of  the  remainder  is  —  30  a^x^. 

Dividing  this  by  twice  the  first  term  of  the  root,  or  6  x^,  we  obtain 
the  second  term  of  the  root,  —  5  a*. 

Adding  this  to  6x2,  we  have  the  complete  divisor,  6x2  —  ba^. 

Multiplying  this  complete  divisor  by  —  5  a^,  and  subtracting  the 
product  from  the  remainder,  there  is  no  remainder. 

Hence,  3  x2  —  5  a^  is  the  required  square  root. 

2.    Find  the  square  root  of 

12ar^  -  22a;3  -f  1  -  20.r^  +  9a;«  +  8a;  +  12x2. 


178 


ALGEBRA. 


Arranging  according  to  the  descending  powers  of  x,  we  have 

9x6  +  12x5-20aj4_22xH12a:2+8ic  +  l     3x3+2x2-4x-l,  Ans. 
9x6 


6x8+2x2 


12x5 
12x5+  4x4 


6x3+4x2-4x 


-24x4 
-24x4-16x3+16x2 


6x3+4x2-8x-l 


6x3- 
6x3- 


4x2 
4x2+8x+l 


It  will  be  observed  that  each  trial  divisor  is  equal  to  the 
preceding  complete  divisor,  ivith  its  last  terin  doubled. 

To  avoid  needless  repetition,  the  last  five  terms  of  the 
first  remainder,  the  last  four  terms  of  the  second,  remainder, 
and  the  last  two  terms  of  the  third  remainder  are  omitted. 

Note.  Since  every  square  root  has  the  double  sign  (§  192),  the 
result  may  be  written  in  a  different  form  by  changing  the  sign  of  each 
term. 

Thus,  in  Ex.  2,  the  answer  may  be  written  1  +  4  x  —  2  x2  —  3  x'**. 

Find  the  square  roots  of  the  following : 

, ,  4.  l-6a-{-na^-6a^-{-a\ 

5.  9a;*-24aj34-4a^  +  16a;+-4. 

6.  20o^-70x+^x*  +  4:9-Sx\ 

7.  a^-^b^-\-c^-2ab-2ac-h2bc. 

8.  9a^  +  l-4a3+-4a«-6a2  +  12a^ 

^*  9.  a;«-4ajV  +  10a^a3+-4fl^a^-20rf  +  25a«. 

10.  9a^  -\-25y^  +  16z^  -\-  SOxy  -24:xz  -  ^Oyz. 

11.  49  m*  —  14  m^n  —  55  mV  +  8  mn^  +  16  7i\ 

12.  49a2-30a3  +  16  +  9a*-40a. 

—    13.   25x'-20i^y-26aff  +  12xf  +  9y', 
14.   16m^4-8mV-23mV-6ma;«  +  9a;«. 


EVOLUTION.  179 

15.  20 ab^-\-9 a' -26d'b^-\-2ob'- 12 a^b. 

16      4 

16.  m2  +  8m  +  12 +  — 2- 

m      Tur 

17.  I_2a5  +  3x2-4a^  +  3a;^-2a75-f  a;«. 

-V    18.    12a;^4-12a;-8af +  94-28a,-2  +  a^  +  10aj3. 

ly.   x^      xy        ^    ^2x^4.a^ 

20    a;^     ■«"     31  a.-^      2.r      4 

■    9      3"^   60         5  "^25* 
21.   4  a«+ 12  a'b  +  25  a*62+  4  a%'-  14  a^ft*-  40  06*+  25  6«. 

-      22.  -  +  _  +  ^^  +  — +  -. 

23.   28a.y  4.  9356  _  15  ^^  -  12 ar^?/  -  8 xy'  -  2a^/  +  16/. 

'    9      3a      Sa'       a'  '^  a' ' 

Find  to  four  terms  the  approximate  square  roots  of: 

^25.   H-4x.  27.   1-x.  29.   a^-^6. 

26.   l4-2a.  28.   l-3a.  ^30.   4:a'-2b. 


SQUARE  ROOT  OF  AN  ARITHMETICAL  NUMBER. 

■  196.   The  square  root  of  100  is  10 ;  of  10000  is  100 ;  etc. 

Hence,  the  square  root  of  a  number  between  1  and  100  is 
between  1  and  10 ;  the  square  root  of  a  number  between  100 
and  10000  is  between  10  and  100 ;  etc. 

That  is,  the  integral  part  of  the  square  root  of  a  number 
of  one  or  two  figures,  contains  one  figure ;  of  a  number  of 
three  or  four  figures,  contains  tivo  figures;  and  so  on. 

Hence,  if  a  j)oint  be  placed  over  every  second  figure  of  any 
integral  number,  beginning  with  the  units'  jjlace,  the  number 
of  points  shows  the  number  of  figures  in  the  integral  part  of 
its  square  root. 


180  ALGEBRA. 


197.   Let  it  be  required  to  find  the  square  root  of  4624 

+  8 
a-\-b 


d'-j-2ab-\-b^  =  4624 
a'  =  3600 


120  +  8 
=  2a-\-b 


go  4-  8  Pointing    the    number    ac- 

cording to  the  rule  of  §  196, 
we  find    that   there   are   two 
1024  =  2ab-\-b      figures  in  the  integral  part  of 
lQ24  its  square  root. 

Let  a  denote  the  greatest 
multiple  of  10  whose  square  is  less  than  4624  ;  this  we  find  by  inspec- 
tion to  be  60. 

Let  h  denote  the  digit  in  the  units'  place  of  the  root ;  then,  the 
given  number  is  denoted  by  (a  +  6)^,  or  a^  +  2ab  -{■  b^. 

Subtracting  a^,  or  3600,  from  4624,  the  remainder  is  1024. 

That  is,  2a6  +  ?)2  =  1024.  (1) 

Since  b^  is  small  in  comparison  with  2  ab,  we  may  obtain  an  ap- 
proximate value  of  b  by  neglecting  the  b'^  term  in  (1). 

Then,         2ab  =  1024,  and  b  =  1^  =  1^  =  8  +. 

2  a       120 

This  suggests  that  the  digit  in  the  units'  place  is  8. 
If  this  be  correct,  2ab  -\-  b^,  or  (2  a  +  b)b,  must  equal  1024. 
Adding  8  to  120,  multiplying  the  sum  by  8,  and  subtracting  the 
product  from  1024,  there  is  no  remainder. 

Hence,  60  +  8,  or  68,  is  the  required  square  root. 

Omitting  the  ciphers,  for  the  sake  of  brevity,  and  con- 
densing the  operation,  it  will  stand  as  follows : 


4624 
36 


68 


128 


1024 
1024 


From  the  above  example,  we  derive  the  following  rule : 

Separate  the  number  into  periods  by  pointing  every  second 
figure,  beginning  with  the  units^  place. 

Find  the  greatest  square  in  the  left-hand  period,  and  write 
its  square  root  as  the  first  figure  of  the  root;  subtract  the 
square  of  the  first  root  figure  from  the  left-hand  period,  and  to 
the  result  annex  the  next  period. 


ji^VOLUTIOy.  181 

Divide  this  remainder,  omitting  tJte  last  figure,  by  twice  the 
j)art  of  the  root  already  found,  and  annex  the  quotient  to  the 
root,  and  also  to  the  trial-divisor. 

Multiply  the  complete  divisor  by  tJie  root-figure  last  obtained, 
and  subtract  the  jyroduct  from  the  remainder. 

If  other  periods  remain,  jyroceed  as  before,  doubling  the  part 
of  the  root  already  found  for  the  next  trial-divisor. 

Note  1.  It  sometimes  happens  that,  on  multiplying  a  complete 
divisor  by  the  figure  of  the  root  last  obtained,  the  product  is  greater 
than  the  remainder. 

In  such  a  case,  the  figure  of  the  root  last  obtained  is  too  great,  and 
one  less  must  be  substituted  for  it. 

Note  2.  If  any  root-figure  is  0,  annex  0  to  the  trial-divisor,  and 
annex  to  the  remainder  the  next  period. 

198.   Required  the  square  root  of  4944.9024. 

1 


We  have,       V4944.9024  =  ^^■ 


^^11:49 


49449024  ^  V49449024 

10000  Vioooo 

49449024  I  7032 


1403 


4490 
4209 


14062 


28124 
28124 


Since  14  is  not  contained  in  4,  we  write  0  as  the  second  root-figure, 
annex  0  to  the  trial-divisor  14,  and  annex  to  the  remainder  the  next 
period,  90.     (See  Note  2,  §  197.) 


7032 


Then,  V4944.9024  =  t2^^  =  70.32. 

100 

The  work  may  be  arranged  as  follows : 

70.32 


4944.9024 

49 

1403 

44  90 

42  09 

14062 

2  8124 

2  8124 

182 


ALGEBRA. 


It  follows  from  the  above  that,  if  a  point  he  placed  over 
every  second  figure  of  any  number,  beginning  with  the  units^ 
place,  and  extending  in  either  direction,  the  ride  of  §  197  may 
be  applied  to  the  result  and  the  decimal  point  inserted  in  its 
proper  position  in  the  root. 


EXAMPLES. 
199.  Find  the  square  roots  of  the  following : 


1.  4225. 

2.  21904. 

3.  508369. 

4.  65.1249. 

5.  .156816. 


6.  .064516. 

7.  3956.41. 

8.  96.4324. 

9.  .00321489. 
10.  12823561. 


11.  75570.01. 

12.  .16216729. 

13.  2666.6896. 

14.  .0062504836. 

15.  86.825124. 


If  there  is  a  final  remainder,  the  number  has  no  exact 
square  root ;  but  we  may  continue  the  operation  by  annex- 
ing periods  of  ciphers,  and  thus  obtain  an  approximate 
root,  correct  to  any  desired  number  of  decimal  places. 

16.   Find  the  square  root  of  12  to  four  decimal  places. 

3.4641 +,    Ans. 


12.00000000 
9 

64 

3  00 
2  56 

686 

4400 
4116 

6924 

28400 
27696 

69281  1  70400 

Find  the  first  five  figures  of  the  square  root  of: 

17.   7.  20.   13.  23.   .2.  26.   .009. 

18/  8.  21.   48.  24.   .056.  27.   .00074. 

19.   10.  22.   64.7.  25.   .39.  28.   SM4:5. 


EVOLUTION.  183 

The  square  root  of  a  fraction  may  be  obtained  by  taking 
the  square  root  of  the  numerator,  and  then  of  the  denomi- 
nator, and  dividing  the  first  result  by  the  second. 

If  the  denominator  is  not  a  perfect  square,  it  is  better  to 
reduce  the  fraction  to  an  equivalent  fraction  whose  denomi- 
nator is  a  perfect  square. 

29.   Find  the  value  of  \/-  to  five  decimal  places. 
Wehave,     ^  =Jl  =  ^=^-^^^^=  .612S7 -.^  Ans. 

Find  the  first  four  figures  of  the  square  root  of: 

30.  ?.  32.  ^.  34.  5.  36.  ^.  38.  ^. 

4  25  5  32  12 


33 
25* 

■^i- 

'-3^ 

1 
2* 

3= J. 

-H 

31.  I  33.1.  35_7.  3,,  15.  gg,  10. 


CUBE  ROOT  OF  A  POLYNOMIAL. 

200.  Since  (a  +  bf  =  a^  +  S  a'b  -f  3  aft^  -f  b%  we  know  that 
the  cube  root  of  a*  -f  3  a^b  +  3  a6^  -f  6*  is  a  +  b. 

It  is  required  to  find  a  process  by  which,  when  the  expres- 
sion a^ -{- S  a^b  +  3  ab^  -{-  b^  is  given,  its  cube  root  may  be 
determined. 

a^^Sa^b  +  Sab^'  +  b^    a-\-b 


3a^-\-3ab-^b' 


3a26  +  3a62  +  63 
3a^b-\-3ab^-{-b^ 


The  first  term  of  the  root,  a,  is  found  by  taking  the  cube  root  of 
the  first  term  of  the  given  expression. 

Subtracting  the  cube  of  a  from  the  given  expression,  the  remainder 
is  Sa^b  +  S  ab'2  +  ¥,  or  (3  a^  +  3  a6  +  b'^)b. 

If  we  divide  the  first  term  of  this  remainder  by  3  a^,  that  is,  by 
three  times  the  square  of  the  first  term  of  the  root,  we  obtain  the 
second  term  of  the  root,  b. 


184  ALGEBRA. 

.  Adding  to  the  trial-divisor  3  a&,  that  is,  three  times  the  product  of 
the  first  term  of  the  root  by  the  second,  and  h^^  that  is,  the  square 
of  the  second  term  of  the  root,  we  obtain  the  complete  divisor, 
3  a2  +  3  a6  +  &2. 

Multiplying  this  by  6,  and  subtracting  the  product,  Za^h+Zah'^-\-h^, 
from  the  remainder,  there  are  no  terms  remaining. 

From  the  above  process,  we  derive  the  following  rule : 

Arrange  the  expression  according  to  the  powers  of  some 
letter. 

Extract  the  cube  root  of  the  first  term,  write  the  result  as  the 
first  term  of  the  root,  and  subtract  its  cube  from  the  given 
expression;  arranging  the  remainder  in  the  same  order  of 
powers  as  the  given  expression. 

Divide  the  first  term  of  the  remainder  by  three  times  the 
square  of  the  first  term  of  the  root,  and  write  the  result  as  the 
next  term  of  the  root. 

Add  to  the  trial-divisor  three  times  the  product  of  the  term 
of  the  root  last  obtained  by  the  part  of  the  root  previously 
found,  and  the  square  of  the  term  of  the  root  last  obtained. 

Multiply  the  complete  divisor  by  the  term  of  the  root  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  terms  remain,  proceed  as  before,  taking  three  times 
the  square  of  the  part  of  the  root  already  found  for  the  next 
trial-divisor. 

EXAMPLES. 
201.  1.   Find  the  cube  root  of 

8x^-36x*y  +  54:x'y'-27f. 


8  ic6  _  36  x^y  +  54  x^y^  -  27  y^ 


2x^-3y,  Ans. 


12x*-lSx^y-{-  9?/2 


-36x4?/  +  54x2?/2-27  2/8 
-36a:^y  +  54a;2y2-27y8 


The  first  term  of  the  root  is  the  cube  root  of  8  x^,  or  2  x^. 
Subtracting  the  cube  of  2ic2,  or  Sx^,  from  the  given  expression, 
the  first  term  of  the  remainder  is  —  36  ic*y. 


EVOLUTION.  185 

Dividing  this  by  three  times  the  square  of  the  first  term  of  the  root, 
or  12  X*,  we  obtain  the  second  term  of  the  root,  —  3  y. 

Adding  to  the  trial-divisor  three  timeg  the  product  of  the  term  of 
the  root  last  obtained  by  the  part  of  the  root  previously  found,  or 
—  18  x'^y,  and  the  square  of  the  term  of  the  root  last  obtained,  or  9  y% 
we  have  the  complete  divisor,  12  x-*  —  ISx^y  -\-  9y^. 

Multiplying  this  complete  divisor  by  —  3  y,  and  subtracting  the 
product  from  the  remainder,  there  is  no  remainder. 

Hence,  2x^  —  Zy  is  the  required  cube  root. 

2.   Find  the  cube  root  of 

28aj3  -  54a;4- a^  +  3a;*  -  9a^  -  27  -  6a:*. 
Arranging  according  to  the  descending  powers  of  x,  we  have 

x6_6a^-f  3a;4+28x3-9x2_54x-27    x2-2x-3. 


3x*-63c8+4x2 


6x^ 
6x6+12x4-  8x8 


3  X*- 12x8+ 12x2 

-  9x2+18x+9 


3x*-12x8+  3x2+18x+9 


-  9x*+36x8 

-  9x4+36x8-9x2-54x-27 


The  second  complete  divisor  Is  formed  as  follows : 

The  trial-divisor  is  three  times  the  square  of  the  part  of  the  root 
already  found  ;  that  is,  3(x2  -  2  x)^,  or  3x*  -  12  x^  +  12x2. 

Three  times  the  product  of  the  term  of  the  root  last  obtained  by  the 
part  of  the  root  previously  found  is  3(  —  3)  (x2  —  2  x),  or  —  9  x2  +  18  x. 

The  square  of  the  term  of  the  root  last  obtained  is  (  —  3)2,  or  9. 

Adding  these,  the  complete  divisor  is  3x*  —  12x8  +  3x2  +  18x  +  9. 

The  last  five  terms  of  the  first  remainder  and  the  last 
three  terms  of  the  second  remainder  are  omitted. 

Find  the  cube  roots  of  the  following : 

4.  l-12a3  +  48a«-64a». 

6.  27m«  +  135m*?i.+  225?MV4-125w8. 

6.  294a62_84a26-343  63  +  8al 

7.  3:f^-6oc^-{-9x^-{-4:a^-9a^~6x-l. 


186  ALGEBRA. 

8.  Sa^ -\-36a'  +  66a'  +  63a^ -\-  SSd" -\-  9a  -{-1. 

9.  30 y'  +  27/  +  12 y  -  4.5y' -  8  -  35  f -\-  21  f, 

*  8       4  "^  6       27' 

11.  9a«-36a  +  a^+21a^-9a^-8-42a2. 

12.  174  a;^  +  8  +  174  a;2  -  60  x'  -  245  x^  +  8  a;^  -  60  x. 

13.  27  a«~  54  a«6  +  63  a'})"-  44  a^^^.^  21  a^ft^-  6  a6^4-  &'• 

14.  6  a.-^^  +  96aj/+  my?f^-  o^  +  24icy  +  6^f  +  96x2/. 

15.  £!-^V^-9  +  ??-^  +  ^. 
27      33  X       :^      x' 

CUBE  ROOT  OF  AN  ARITHMETICAL  NUMBER. 

202.  The  cube  root  of  1000  is  10 ;  of  1000000  is  100 ;  etc. 
Hence,  the  cube  root  of  a  number  between  1  and  1000  is 

between  1  and  10 ;  the  cube  root  of  a  number  between  1000 
and  1000000  is  between  10  and  100;  etc. 

That  is,  the  integral  part  of  the  cube  root  of  a  number  of 
one,  two,  or  three  figures,  contains  one  figure ;  of  a  number 
of  four,  five,  or  six  figures,  contains  two  figures ;  and  so  on. 

Hence,  if  a  point  he  placed  over  every  third  figure  of  any 
integral  number,  beginning  with  the  units'  ijlace,  the  number 
of  points  shows  the  number  of  figures  m  the  iyitegral  part  of  its 
cube  root. 

203.  Let  it  be  required  to  find  the  cube  root  of  157464. 


a^^3a%  +  3  ab^  +  W=  157464 
a^  = 125000 


50  +  4  =  a  +  6 


3a2=7500 

3ab=:    600 

b'=     16 

3a2-h3a6  +  62  =  8116 


32^64:  =  3a'b-\-3ab'-\-b^ 


32464 


Pointing  the  number  according  to  the  rule  of  §  202,  we  find  that 
there  are  two  figures  in  the  integral  part  of  its  cube  root. 


EVOLUTION.  187 

Let  a  denote  the  greatest  multiple  of  10  whose  cube  is  less  than 
157464 ;  this  we  find,  by  inspection,  to  be  50. 

Let  b  denote  the  digit  in  the  units'  place  of  the  root ;  then,  the 
given  number  is  denoted  by  (a  +  6)^,  or  a^  +  3  a^b  +  3  ab^  +  b^. 

Subtracting  a^  or  125000,  from  157464,  the  remainder  is  32464. 

That  is,  3  a^b  +  3  aft^  +  53  =  32464.  (1) 

Since  3  ab^  and  b^  are  small  in  comparison  with  3  a^b,  we  may 
obtain  an  approximate  value  of  b  by  neglecting  the  Zab'^  and  b^ 
terms  in  (1). 

Then,        3  a%  =  32464,  and  6  =  §^464  ^  32464  ^ 

3a2        7500 

This  suggests  that  the  digit  in  the  units'  place  is  4. 

If  this  be  correct,  3a^b  +  Sab^-{-  b\  or  (Sa^  +  3a6  -f  b^)b,  must 
equal  32464. 

Adding  to  7500  3  a6,  or  600,  and  62,  or  16,  the  sum  is  8116  ;  multi- 
plying this  by  4,  and  subtracting  the  product  from  32464,  there  is  no 
remainder. 

Hence,  50  +  4,  or  54,  is  the  required  cube  root. 

Omitting  the  ciphers  for  the  sake  of  brevity,  and  con- 
densing the  operation,  it  will  stand  as  follows : 

54. 


157464 
125 

7500 

600 

16 

8116 

32464 
32464 

From  the  above  example,  we  derive  the  following  rule : 

Sejmrate  the  number  into  periods  by  pointing  every  third 
Jigu7^e,  beginning  with  the  units'  place. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write  its 
cube  root  as  the  first  figure  of  the  root;  subtract  the  cube  of  the 
first  root-figure  from  the  left-hand  period,  and  to  the  result 
annex  the  next  period. 

Divide  this  remainder  by  three  times  the  square  of  the  part 
of  the  root  already  found,  ivith  two  ciphers  annexed,  and  write 
the  quotient  as  the  next  figure  of  the  root. 


188  ALGEBRA. 

Add  to  the  trial-divisor  three  times  the  product  of  the  last 
root-figure  by  the  ^yart  of  the  root  previously  found,  with  one 
cipher  annexed,  and  the  square  of  the  last  root-figure. 

Multiply  the  complete  divisor  by  the  figure  of  the  root  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  periods  remain,  proceed  as  before,  taking  three 
times  the  square  of  the  part  of  the  root  already  found,  with  two 
ciphers  annexed,  as  the  next  trial-divisor. 

Note  1.     Note  1,  p.  181,  applies  with  equal  force  to  the  above  rule. 

Note  2.  If  any  root-figure  is  0,  annex  two  ciphers  to  the  trial- 
divisor,  and  annex  to  the  remainder  the  next  period. 

204.  If,  in  the  example  of  §  203,  there  had  been  more 
periods  in  the  given  number,  the  next  trial-divisor  would 
have  been  three  times  the  square  of  a +5,  or  3a^+6a6-h36^. 

We  observe  that  this  may  be  obtained  from  the  preceding 
complete  divisor,  3  a^  -f  3  a6  +  b^,  by  adding  to  it  its  second 
term,  3  ab,  and  twice  its  third  term,  2  W. 

Hence,  if  the  first  number  and  twice  the  second  number 
required  to  complete  any  trial-divisor,  be  added  to  the  comr 
plete  divisor,  the  result,  with  two  ciphers  annexed,  will  be  the 
next  trial-divisor. 

205.  Required  the  cube  root  of  8144.865728. 


We  have,   ^8144.865728  =  ^^^^^^^^^^^  =  ^^1^^^^^^^^. 
^    1000000  ^1000000 


8144865728  1 

8        1 

120000 

144865 

600 

1 

120601 

120601 

600 

24264728 

2 

12120300 

12060 

4 

1213236 

4 

24264728 

2012 


EVOLUTION. 


189 


Since  1200  is  not  contained  in  144,  the  second  root-figure  is  0 ;  we 
then  annex  two  ciphers  to  the  trial-divisor  1200,  and  annex  to  the 
remainder  the  next  period,  865. 

The  second  trial-divisor  is  formed  by  the  rule  of  §  204.  Adding  to 
the  complete  divisor  120601  the  first  number,  600,  and  twice  the  second 
number,  2,  required  to  complete  the  trial-divisor  120000,  we  have  121203  ; 
annexing  two  ciphers  to  this,  the  result  is  12120300. 

Then,  \/8144. 865728  =?^  =  20.12. 

100 

The  work  may  be  arranged  as  follows : 

20.12 


8144.865728 
8 

120000 

600 

1 

120601 

144  865 
120  601 

600 
2 

24  264728 

12120300 

12060 

4 

12132364 

24  264728 

It  follows  from  the  above  that,  if  a  point  he  placed  over 
eve^-y  third  figure  of  any  number ,  beginning  with  the  units' 
place,  and  extending  in  either  direction,  the  rule  of  §  203 
may  be  applied  to  the  result,  and  the  decimal  point  inserted 
in  its  proper  position  in  the  root. 


EXAMPLES. 
206.   Find  the  cnbe  roots  of  the  following : 


1.  19683. 

2.  148877. 

3.  59.319. 

4.  .614125. 

5.  2515456. 


6.  857.375. 

7.  .224755712. 

8.  46.268279. 

9.  523606616. 
10.  187149.248. 


11.  .000111284641. 

12.  788889.024. 

13.  444.194947. 

14.  338608873. 

15.  .001151022592. 


190  ALGEBRA. 

Find  the  first  four  figures  of  the  cube  root  of : 

16.  3.  18.   9.1.  20.   ^.  22.   -. 

8  9 

17.  7.  19.   .02.  21.   ii.  23.  -• 

27  5 

207.  If  the  index  of  the  required  root  is  the  product  of 
two  or  more  numbers,  we  may  obtain  the  result  by  successive 
extractions  of  the  simpler  roots. 

For  by  §  189,  fVa)'""  =  a. 

Taking  the  nth  root  of  both  members, 

C-Var=^a.  (1) 

Taking  the  mth  root  of  both  members  of  (1), 

mn/—  ^/  n/— 

■\a=  V  S/a. 

Hence,  the  mnth  root  of  an  expression  is  equal  to  the  mth 
root  of  the  nth.  root  of  the  expression. 

Thus,  the  fourth  root  is  the  square  root  of  the  square 
root ;  the  sixth  root  is  the  cube  root  of  the  square  root,  etc. 

EXAMPLES. 
Find  the  fourth  roots  of  the  following : 

1.  81  a^  +  216  a^y  +  216  a^ft^  +  96  aW  +  16  h\ 

2.  l-12aj  +  50a^-72a^-21a;*4-72a^+50aj6+12aj^+a^. 

3.  16  a«-32  a^-40  a«+88  a'-\-  49  a^-88  a^-4:0  a2+32  a+16. 

Find  the  sixth  roots  of  the  following  : 

4.  ici«  4-  6  x^Y  +  15  ^'y  +  20  ^f  +  15  a;y  +  6  ^tf  +  y'^ 

5.  a«-12a^  +  60a^-160a3  +  240a2-192a4-64. 

6.  Find  the  fourth  root  of  209727.3616. 

7.  Find  the  sixth  root  of  .009474296896. 


THEORY  OF   EXPONENTS.  191 


XX.    THEORY  OF  EXPONENTS. 

208.  In  tlie  preceding  chapters,  an  exponent  has  been 
considered  only  as  a  positive  integer. 

Thus,  if  m  is  a  positive  integer, 

aJ^  =  axaxay.  •  •  •  to  m  factors.        (§  6) 

209.  Let  m  and  n  be  positive  integers. 
Then,  «"'  =  axaxax  •••to  m  factors, 

and  a""  =  a  K  a  xax  •  •  •  to  n  factors. 

Whence,  cr  x  a'*  =  a  x  a  x  a  x  •  •  •  to  m  -\-  n  factor^. 

That  is,    or  x  a"  =  a'"+".  (1) 

This  proves  the  law  stated  in  §  46  for  all  positive  integral 
values  of  the  exponents. 

Again,  (a*")"  =  a*"  x  a"*  X  a"*  x  •  •  •  to  n  factors 

^m+m+m+—to  n  terms 

That  is,         {ory  =  a""*.  (2) 

This  proves  the  first  paragraph  of  the  law  stated  in  §  186 
for  all  positive  integral  values  of  the  exponents. 

210.  It  is  found  convenient  to  employ  exponents  which 
are  not  positive  integers;  and  we  proceed  to  define  them, 
and  to  prove  the  rules  for  their  use. 

It  will  be  convenient  to  have  all  forms  of  exponents  sub- 
ject to  the  same  laws  in  regard  to  multiplication,  division, 
etc. ;  and  we  shall  therefore  find  what  meanings  must  be 
attached  to  fractional,  negative,  and  zero  exponents  in  order 
that  equation  (1),  §  209,  may  hold  for  all  values  of  m  and  n. 


192  ALGEBRA. 

211.  Meaning  of  a  Fractional  Exponent. 

1.  Required  the  meaning  of  aj^ 

If  (1),  §  209,  is  to  hold  for  all  values  of  m  and  n,  we  have 

5  5  5  5  I  5  ,5 

Hence,  a^  is  such  an  expression  that  its  third  power  is  a^. 

Then/  a^  must  be  the  cube  root  of  a^ ;  or,  a^  =  -v/o^. 

p 

2.  Required  the  meaning  of  a',  where  p  and  q  are  any 

positive  integers. 

If  (1),  §  209,  is  to  hold  for  all  values  of  m  and  n,  we  have 

p       p       p  p  p  p  p 

a^  xa^  X  a^  X  •••  to  g  factors  =  a^'^^'^^'^'"^''^^""''=  a^"""  =  a^. 

p 
Hence,  a^  is  such  an  expression  that  its  qth  power  is  a^ 

p  p  

Then,  a*  must  be  the  gth  root  of  a^ ;  or,  a'  =  va^. 

Hence,  in  a  fractional  exponent,  the  numerator  denotes  a 
power,  a7id  the  denominator  a  root. 

For  example,  a^  =  -\/'a^;  6^  =  V6*;  x^  =  -y/x;  etc. 

EXAMPLES. 

212.  Express  the  following  with  radical  signs  ; 

1.   a^.       3.   4.x\  ^-5.   aM.    ^7.   6x^yK         9.   abh'd'^. 

m    1  q 

_  2.   b\       4.   9  abK       6.   m%i       8.   8  aW.  ^0.   3  x^^^:^^. 

Express  the  following  with  fractional  exponents : 

11.  v^3.  13.    Vm^  15.   2-^«.  17.    ^/^^/6"^ 

12.  </x.  14.   ^6^.  16.   5^.  18.  .^/m'</^'. 

19.    l-Vx-y/f.  20.    V^VftVc^. 


THEORY  OF   EXPONENTS.  193 

213.  Meaning  of  a  Zero  Exponent. 

If  (1),  §  1?09,  is  to  hold  for  all  values  of  m  and  n,  we  have 

or  X  a"  =  «"*+"  =  a"*. 

Whence,  a«  =  — =  1. 

a*" 

Hence,  ^/ie  zero  power  of  any  quantity  is  equal  to  1. 

214.  Meaning  of  a  Negative  Exponent. 

1.  Required  the  meaning  of  cr'-K 

If  (1),  §  209,  is  to  hold  for  all  values  of  m  and  n,  we  have 

a-3  xa^  =  a-3+3  =  a'  =  1.  (§  213) 

Whence,  a~^  =  — 

2.  Required  the  meaning  of  a  *,  where  s  is  a  positive 
integer  or  a  positive  fraction. 

If  (1),  §  209,  is  to  hold  for  all  values  of  m  and  n,  we  have 

a-'  xa'  =  a  '+'  =  a''=l.  (§213) 

Whence, 


a 

» -*- 

~a'' 

L 

a"^  = 

1 
at' 

3  a;- 

■y 

4  = 

3 

I 

215.   It  follows  from  §  214  that 

Any  factor  of  the  numerator  of  a  fraction  may  he  trans- 
ferred to  the  denominator,  or  any  factor  of  the  denominator 
to  the  numerator,  if  the  sign  of  its  exponent  he  changed. 

Thus,  ^=-*l-  =  ^^'  =  ^%tc. 


194  ALGEBRA. 

EXAMPLES. 
216.   Write  the  following  with  positive  exponents : 
1.   o}h-\  5.   a-^x-\  9.    6aj-V^. 

3.  a^V*.  7.   4a-«?>~l  11.    6m-2n"V- 

4.  2aV9.  8.   a"^6-V.  ^12.   cj-^^-Vto. 

Transfer  the  literal  factors  from  the  denominators  to  the 
numerators  in  the  following : 

1/.    — i».  „ — 

18.    -iCl_.       20.    ^^lM- 
2/3^  7  m-^a?^  4  h'^c  8  6~^?/^' 

Transfer  the  literal  factors  from  the  numerators  to  the 

denominators  in  the  following : 

/^ 

21.    — .  23.    ^-  25.    '^^.        27.    ^^l!^. 

2^  5  x^2/~ 


13. 

1 

15. 

2 

14. 

a;-t 

16. 

1 

2 

3 

6* 

24. 

6 

22.    ^.  24.    ^^i^.         26.   ^l!^.       28.    ^^"'^l 

2/"*  96-^71  3 

217.  Since  the  definitions  of  fractional,  negative,  and  zero 
exponents  were  obtained  on  the  supposition  that  equation 
(1),  §  209,  was  to  hold  universally,  we  have  for  all  values  of 

m  and  n 

or  xa''  =  «"*+". 

For  example,   a^  x  a~^  =  o?~^  =  a~^ ; 
a^  X  a~^  =  a^~^  =  a^^ ; 
a  X  Va*  =  a  X  a^  =  a^^^  =  a^  •  etc. 


THEORY  OF   EXPONENTS.  195 


EXAMPLES. 

Find  the  values  of  the  following : 


1. 

a'  X  a-\ 

4. 

m^ 

X  m  i             7.   5  ic-2  X  4  a;  i 

2. 

o?  X  a-^. 

5. 

2  J 

■  X  n"i             8.    m^  X  ^/m. 

3. 

x-^  X  x-^. 

10.    n-^x-^ 

n  ^ 

6. 

a  X 

3a-V             9.   c^x^y^^. 

13.  £c-«/  X  ^x'y-\ 

14.  m^w"^  X  \  m-*n~K 

11.  iVa'x^ 

12.  3a\/Fx 

15.       ^     X  a-'x-K 
a-'x^ 

16.   Multiply  a  +  2  a^  -  3  a^  by  2  -  4  a"^  -  6  a"i 
2  _4a"^-    6a~^ 


2a  +  4a^    -   6a^ 


2  a  -20a^  +  18a"^ 


Ans. 


Note.     It  must  be  carefully  borne  in  mind,  in  examples  like  the 
above,  that  the  zero  power  of  any  quantity  is  equal  to  1  (§  213). 

Multiply  the  following : 

17.  J  +  ah^  4-  b^  by  a^  -bK 

18.  4a;"^ -6a;"^  + 9  by  2a;~^  + 3. 

19.  2a-i-7-3a  by  4a-i4-5. 

20.  x~^  +  2  a;"^  +  4  rc"^  4-  8  by  x~^  -  2. 

21.  x^  -{■  xhj^  -\-  y^  by  x^  —  x^y^  +  y^- 

22.  m  —  2  m^w"^  +  m^n~^  by  m^n~^  —  2  m^w"^  +  w"». 


196 


ALGEBRA. 


23.  a~25-3  _^  ^-3^-4  _  ^-4^-5    by    ^-15-2  _  ^-2^-3  _  ^-35-4_ 

24.  m-^4-  2  m"3n-i+  3  ??r^n-2  by  2  /?r^—  4  n"^  +  6  m^-^. 

25.  2  ah-'  +  a^  -  4  ^-^6^  by  2  a^  -  5^  _  4  ^-^54 

01  21  12  11  ^ 

26.  3m%3_477ia;3-f-m^ic  by  6m^ic~3  + 8m~%~3-|-2m~*. 


218.    To  prove  —  =  a*^""  for  all  values  of  m  and  n. 


By  §  215, 


-  =  a*"  X  a-"  =  a*"-",  by  (1),  §  209. 


For  example, 


-I. 


a 


a" 


=  2Ll  =  a-^i 


a"'^;  etc. 


EXAMPLES. 


Divide  the  following : 

1.  a^  by  a\ 

2.  ic  by  x^. 

3.  m^  by  ?)i~». 

2  1 

4.  a  3  by  a^. 

5.  6-2  by  V6^. 


6.   2Vx  by  ic" 
1 


7.    n^  by 


^n^ 


8.    lOa-^6-^  by  5  a^h'\ 


9.   6 


by  2</x- 


10.    Divide  2  a*  -  20  +  18  a~*  by  a  +  2  a*  -  3  a^. 


2a3 -20  +  18a  ^ 
2  a^  +  4  a^  -  6 


a  +  2 


a3  -SOS'S' 


2  a" 


4  a  ^  -  6  a-i,  ^ws. 


_4a^-14  +  18a"^ 
_4«i_    8  +  12a~^ 


6-12a"^  +  18a~^ 


-   6-12a"^  +  18a 


-f 


THEORY   OF   EXPONENTS.  197 

Note.     It  is  important  to  arrange  the  dividend,  divisor,  and  each 
remainder  in  the  same  order  of  powers  of  some  common  letter. 

. .  ^7 

Divide ^e  following: 

11.   a^  -f  h"-  by  a^  -f  ftl         12.   a-^  -  1  by  a^  -  1. 

13.  ic4  _  2  _|_  a;-4  ]3y  ^  _^  2  +  x-'^. 

14.  a-4a^-f  6a*-4a^-f-l  by  a*-2a*-hl. 

16.  a;^  —  3  aJ2/^  +  3  ^y^  —  ?/  by  a;*  —  y^. 

.     16.  m~^  —  3  m"^  —  4  m"^  by  m"^  -f-  2  m~^. 

17.  9  ic^^^^  _^  5  +  a;-V"'  by  3  ^-^  -  x'Y'  +  a;-^'- 

18.  a~hn  —  5  am~^  +  4  a^wi~^  by  a~^m*  —  a~^m  —  .2  a"\. 

19.  ab-^  _  10  6^  +  9  a-'b  by  a*  +  2  6^  -  3  a'M. 

20.  ?7i^  —  2  a;^  +  m'^a^  by  ??i~^a;*  —  2  m'^x^  +  m-^ici 

219.    To  prove  (a'")"  =  a""*  /o?-  all  values  of  m  and  n. 
We  will  consider  three  cases,  in  each  of  which  m  may 
have  any  value,  positive  or  negative,  integral  or  fractional. 

I.  Let  n  be  a  positive  integer. 

The  proof  of  (2),  §  209,  holds  if  w  is  a  positive  integer, 
whatever  the  value  of  m. 

P 

II.  Let  n  =  -,  where  p  and  q  are  positive  integers. 

Then,  by  the  definition  of  §  211, 


(ay  =  V(a"'y  =  -Var^  (§  219,  I.)  =  a « . 
III.   Let  n  =  —s,  where  s  is  a  positive  number. 
Then,  by  the  definition  of  §  214, 

(ay  =  -i-  =  —  (§  219,  I.  or  II.)  =  a-'. 
^     ^         (ay     a""^  '  ^ 

Therefore,  the  equation  holds  for  all  values  of  7n  and  n. 


ti 


198  ALGEBRA. 

For  example,  ((X^)-^  =  a^x-^  =  a-^^^j 

(V^)t  =  (a^)t  =  a^xf  =  a* ;  etc. 

EXAMPLES.  V 

220.  Find  the  values  of  the  following : 

1.  {a')-\  7.    (x-'^f.  12.    (\/^-^.      ^  ^^ 

2.  (a-)^  8.    {^^)K  13.   (-^y.    .'''c>^ 

3.  {x^f.        ■  9.   (a-)l  t^ 

4-    (--¥.  10.    (^)e  '''    ^^  ">"  ^  , 

4  11      f  -  1  !?-i  -!!L 

6.    (a-^"^.  '   W    *  16.    (a;-    )"--. 

221.  The  value  of  a  numerical  quantity  affected  with  a 
fractional  exponent  may  be  found  by  first,  if  possible,  ex- 
tracting the  root  indicated  by  the  denominator,  and  then 
raising  the  result  to  the  power  indicated  by  the  numerator. 

1.  Find  the  value  of  (-  8)1 

We  have, 

(-  8)^  =  [(-  8)^]2  =  (v/38)2  =  (_  2)2  =  4,  Ans. 

EXAMPLES. 
Find  the  values  of  the  following : 

2.  25l  6.  49"^  10.  16-1  14.  32-1 

3.  9l  7.  (-27)-^  11.  (-32)1  15.  (-64)1 

4.  8l  8.  4i  12.  64l  16.  (-243)^ 

5.  Sll  9.  343l  13.  (-125)i  17.  (-128)-^ 


U' 


THEORY  OF  EXPONEXTS.  199 

222.  To  j)rove  {ahy  =  al'h'^  for  any  value  ofn. 

I.  Let  n  be  a  positive  integer. 

Then,  (aby  =  ab  x  ab  x  ab  x  -"  to  n  factors 

=  (a  X  a  X  •  •  •  to  71  f actors)(6  x  &  X  •  •  •  to  n  factors) 
=  a"6". 

V) 

II.  Let  n  =  -,  where  p  and  q  are  positive  integers. 

Then  by  §  219,     [(aft)']'  =  (aby  =  a^b^,  by  §  222,  I. 

And  by  §  222,  I.,  [a^b^y  =  (Jy(b^y  =  a^b^. 

Therefore,  [(a^)']'  =  [a'6']'. 

Taking  the  ^th  root  of  both  members,  we  have 

p       p  p 
{aby  =  a^¥. 

III.  Let  w  =  —  s,  where  s  is  any  positive  number. 

Then,  (a6)-=-i-=:  J-  (§  222,  L  or  IL)=a-6-. 
^        {aby     a'b'  ^         '  ^ 

MISCELLANEOUS    EXAMPLES. 

223.  Square  the  following  by  the  rule  of  §  78  or  §  79 : 

1.   2a?  +  36i      2.   5x-Y-2a^y-'.       3.   3aV^-4a"V. 
Extract  t'he  square  roots  of  the  following : 
4.   a-*bi       5.   25mn-U       6.   ^^.      7.   ^!^. 

8.  4a*  +  4a*-19-10a-^  +  25a-i 

9.  9a;"t_12a;-2  +  10a;-^-4a;"^4-l. 

10.   a'b-^  -  8  ah''  +  10  a'b-'  +  24  ah''  +  9  ab-\ 


200  ALGEBRA. 

Extract  the  cube  roots  of  the  following : 

11.   a^h-\      12.   xy^z-^.      13.    -27mV^.       14.   ^^. 

15.  8a4-12aV^  +  6aV3-6-^. 

16.  x-^  +  6a;-*  +  3a;*  -  28aj^  -  9a;^  +  54a;^  -  27a;l 

Simplify  the  following : 

^2»»— 3n    sy    «— Sm-n  1        1 

17.  ^-^^ 20.   lirf-^f^K 

1  1      1  _*_  ae-y  x+y 

19.    (a;^i  X  a;^)".  22.    (a=^+^^a  ^  )  "  . 

23.  (2"+-^  -  2  X  2«)  X  (2-2  X  2-"-2). 

24.  (a;^r"'^^^(|y- 

25.  (c.*-a*y+(l  +  a^*a*y  ^V 
1  +  a*a;*(a~*a;*  -  a^x~^  -  a^x^)                      ^ 

26.  A±l_4._£zil_.  28.    a;-^  - 2/"^     ^~^  +  ^"1 
a;i_2/i     a;^H-2/^  '    a;"*  +  2/"*      a;"*  -  2/~* 

3n  _3»  3n 

27.  ^'-^  '  29.   ^^-1     a^"-l. 

n  M  n  n 

30.  ^'  +  y-\^-'-y-'_x, 

31     <^^  +  2  6^     a^-2a^6^H-4  6^ 
a^  -2h^     a*  +  2  a^  6*  +  4  6^ 


.^^ 


RADICALS.  201 


XXI.  RADICALS. 

224.  A  Radical  is  a  root  of  an  expression,  indicated  by  a 
radical  sign ;  as  Va,  or  V»  +  1. 

If  the  indicated  root  can  be  exactly  obtained,  the  radical 
is  called  a  rational  quantity ;  if  it  cannot  be  exactly  obtained, 
it  is  called  an  irrational  quantity,  or  surd. 

225.  The  degree  of  a  radical  is  denoted  by  the  index  of 
the  radical  sign ;  thus,  -\/x  +  1  is  of  the  third  degree. 

226.  Most  problems  in  radicals  depend  for  their  solution 
on  the  following  principle : 

111 
For  any  value  of  n,     (abf  =  a"  x  6«.  (§  222) 

That  is,  -\/ab  =  -s/d  x  ^/b. 

REDUCTION  OF  A  RADICAL  TO  ITS  SIMPLEST  FORM. 

227.  A  radical  is  said  to  be  in  its  simplest  form  when  the 
expression  under  the  radical  sign  is  integral,  is  not  a  perfect 
power  of  the  degree  denoted  by  any  factor  of  the  index  of 
the  radical,  and  has  no  factor  which  is  a  perfect  power  of 
the  same  degree  as  the  radical. 

228.  Case  I.  When  the  expression  under  the  radical  sign 
is  a  j)erfect  power  of  the  degree  denoted  by  a  factor  of  the  index. 

1.  Reduce  VS  to  its  simplest  form. 

We  have,  v/8  =  \^  =  2^  (§  211)  =  2^  =  \^,  Ans. 

EXAMPLES. 
Reduce  the  following  to  their  simplest  forms : 

2.  -im.  3.    -^.  4.    ^25.  5.    ^^64. 


m 

^4. 

9. 
10. 
11. 

^216. 

^100. 
^243. 

ALGEBRA. 

15. 
16. 
17. 

6. 

12.    </121  a'b\ 

v'Sl  m»ni2 

7. 

13.    ^125  a;^^^ 

V^SorV. 

8. 

14.    ^32  a''m'. 

-\/27  a«a^. 

229.  Case  II.  When  the  expression  under  the  radical  sign 
is  integral,  and  has  a  factor  which  is  a  perfect  power  of  the 
same  degree  as  the  radical. 

1.   Eeduce  V54  to  its  simplest  form. 

We  have,  \/U  =  </2rx2  =  V21  x  </2  (§  226)  =  3  v^,  Ans. 


2.   Reduce  V3  a^b  -  12  a^b^  +  12  ab^  to  its  simplest  form. 


V3a36_i2a262  +  i2a&8  =  V(a2  _  4 a6  +  4 62)3  a6 

=  Va^-4ab  -\-4:bW3ab  =(a-2b) VSab,  Ans. 

From  the  above  examples,  we  derive  the  following  rule : 
Resolve  the  expression  under  the  radical  sign  iyito  tico  fac- 
tors, the  seco7id  of  which  contains  no  factor  which  is  a,  perfect 
power  of  the  same  degree  as  the  radical. 

Extract  the  required  root  of  the  first  factor,  and  prefix  the 
result  to  the  indicated  root  of  the  second. 


EXAMPLES. 
Reduce  the  following  to  their  simplest  forms : 


3.    V28.       7.   3V98.        11.    ^375.  15.    Vl92mV. 


4.  V99.       8.    V150.        12.    a/162.  16.    </12Sxf^. 

5.  V80.       9.   5^108.      13.    a/128.  17.    </^lM\ 


6.    a/40.      10.    V243.        14.    V242aV.      18.    V96a^6V. 

19.    Vi08^«T72^^.         21.    V(a^-4  62)(a-2  6). 


20.    a/135  a^2/^- 108  a^/.       22.    A/5ar''  +  30a^  +  45a;. 
23.    a/27  a^b  -  36  o:'W  + 12^^. 


24.    -\/{^-x-Q){x'-\-2x-15). 


RADICALS.     .  203 

If  the  expression  under  the  radical  sign  has  a  numerical 
factor  which  cannot  be  readily  factored  by  inspection,  it  is 
convenient  to*i'esolve  it  into  its  prime  factors. 


25.  Reduce  vl944  to  its  simplest  form. 

\/l944  =  y/WVW>  =  </2»  X  38  x^  =  2x3x'^^  =  6v^,  Ans. 

26.  Reduce  V125  x  147  to  its  simplest  form. 
V'l25xl47  =  V68x3x72=V52x72 x  V3xb  =  3x 7 x \/l5=35>/l5,  Ans. 

Reduce  the  following  to  their  simplest  forms : 

27.  VSei.  30.    V125  X  135.        33.    -J/4ll6. 

28.  y/2625.        31.    V98  x  336.  34.    ^196  x  392. 

29.  V3528.         32.    ^/1125.  35.    -v/40  x  45  x  48. 

36.    V75a=^xl05a6  x  189  61 

230.  Case  III.  When  the  expression  under  the  radical 
sign  is  a  fraction. 

In  this  case,  the  radical  may  be  reduced  to  its  simplest 
form  by  multiplying  both  terms  of  the  fraction  by  such  an 
expression  as  will  make  the  denominator  a  jierfect  jyotver  of  the 
same  degree  as  the  radiccd,  and  then  proceeding  as  in  §  229. 

1.     Reduce  \-^—^  to  its  simplest  form. 

'Oft 

Multiplying  both  terms  of  the  fraction  by  2  a,  we  have 

/X  =  J9xI^^J_0_x2«  =  JIIIxV2^=^V2^,^ns. 

EXAMPLES. 
Reduce  the  following  to  their  simplest  forms : 

'■4     ^-4     -4-    '-4 


204  •     ALGEBRA, 


W|  "^  "4  ^'-^'^■ 

'a/I  "€  "€  "x'ff- 

8.  #  12.  4  16.  Jl.  »  Jf. 

9.  /i-  13.  4  17.  V^.  21. 


3  21       3/50^ 

10  or^'  '    ^16f' 


22      -  f"-  -r  ^  o«5         -      .  /2  a^  -  8  «  +  8 


I.    J^H.  23.    ^^  J 


231.  To  Introduce  the  Coefficient  of  a  Radical  under  the 
Radical  Sign. 

The  coefficient  of  a  radical  may  be  introduced  under  the 
radical  sign  by  raising  it  to  the  power  denoted  by  the  index. 

1.  Introduce  the  coefficient  of  2aV3x^  under  the  radical 
sign.  ,  

2  aVSx^  =  \/8a3^3x2  =  y/Sa^  xS x'^  (§  226)  =  v/24 a^x'^,  Ans. 

Note.  A  rational  quantity  may  be  expressed  in  the  form  of  a 
radical  by  raising  it  to  the  power  denoted  by  the  index,  and  writing 
the  result  under  the  corresponding  radical  sign.  . 

EXAMPLES. 
Introduce  under  the  radical  signs  the  coefficients  of : 


2. 

5V2. 

5.   5^/4. 

8.   4aV8a. 

11.    ^f^^y\ 

3. 

8V3. 

6.   2^5.- 

9.   7x'^/6a^. 

12.   3m2</2m. 

4. 

4^/6. 

7.   3^2. 

10.   3ab^5a'. 

13.   2aA/7V, 

14.   (l  +  awS.  16.   ^J!i± 

15.  (»-i)V^+i.    17.  j^vr 


2x 


232.   Similar  radicals  are  radicals  which  do  not  diifer  at 
all,  or  differ  only  in  their  coefficients ;  as  2Vax^  and  3  Vaa^. 


RADICALS.  205 

ADDITION  AND  SUBTRACTION  OF   RADICALS. 

233.  To  add  or  subtract  similar  radicals  (§  232),  add  or 
subtract  their  coefficients,  and  prefix  the  result  to  their 
common  radical  part. 

1.  Required  the  sum  of  V^  and  V45. 
Reducing  each  radical  to  its  simplest  form.(§  229),  we  have 

V20  +  V45  =  \/4x~5  +  y/WVl  =  2 V5  +  3 V5  =  5 V5,  Ans. 

2.  Simplify  ^  +  ^-^. 


18 

=  ^  V2  +  ^  V6  -  f  V2  =  ^  V6  -  J  v^,  Ans. 

Rule. 

Reduce  each  radical  to  its  simplest  form. 
Unite  the  similar  radicals,  and  indicate  the  addition  or  sub- 
traction of  those  which  are  not  similar. 

EXAMPLES. 
Simplify  the  following : 

3.  V75  +  V12.       5.    VSO-VlSO.       7.    ^192-^3. 

4.  V98-V18.      6.    </U-{-Vl6.        8.    \/32-</l62. 
9.    V27+V108-V48.         10.    Vi75  -  V  ll2  -  V44. 

„,|,^.       ,.4.4        ,3.J,^. 

14.    V5+V245-V320.         16.    </5  -  ^/320 -{- </72. 


206  ALGEBRA. 

18.  bWS  a'b  +  ab  V50  a%^  -  aVl28  ab'. 

19.  m2-C/32^2^m^T08^  +  ^500^. 


20.    V50  a^  -  75  a^o;  _  V32  a  V  _  48  a^, 

23.  .^-^24-^^+^375. 

24.  .{/243-</48- -1/768. 

25.  V32-V72+V125+V162-V500. 

26.  a^Vl50^  +  V96l^-V54^-a5V24^. 


27.    V63  a%  +  6  V160  aft^  -  VWab"  -  a  V252  5. 


«^  M-4.*4-4 


30.  V80a;^  +  40a^  +  5a;  +  V45aj3-60x2  +  20a?. 

31.  2Vl2a;2_^6Q3,2^_|_75^2_^48^^_72a;2/  +  27?/2. 

2a 


32 


-^ 
+  & 


52 


Va2- 


TO    REDUCE    RADICALS    OF    DIFFERENT    DEGREES    TC 
EQUIVALENT  RADICALS  OF  THE  SAME  DEGREE. 

234.   1.   Reduce  V2,  V3,  and    V5   to  equivalent  radi- 
cals of  the  same  degree. 

By  §  211,  V2  =  2^  =  2T^  =  ^/2^  =  ^/6i  ; 


RADICALS.  207 

We  then  have  the  following  rule : 

Express  the  radicals  ivith  fractional  exponents,  and  reduce 
these  exponents  to  a  common  denominator. 

Note.     The  relative  magnitude  of  radicals  may  be  determined  by 
reducing  them,  if  necessary,  to  radicals  of  the  same  degree. 

Thus,  in  Ex.  1,  v^l25  is  greater  than  y/^,  and  v^81  than  \/64. 
Hence,  -Vb  is  greater  than  v^,  and  v^  than  V2. 

EXAMPLES. 
Reduce  to  equivalent  radicals  of  the  same  degree : 

2.  V3  and  V5.  7.  y/xy,  ^yz,  and  yfzx. 

3.  V2  and  -v^.  8.  -s/^a,  ^/2b,  and  -^. 

4.  -V^  and  ^/o^.  9.  's/2,  -5/8,  and  ^/TS. 

5.  V2  and  ^12.  10.  \T^  and  ^/TT^. 

6.  ^  and  </6.  11.  V^^+^  and  ^^^^. 

12.  Which  is  the  greater,  ^  ov  ^? 

13.  Which  is  the  greater,  Vll  or  V5  ? 

14.  Which  is  the  greater,  VlO  or  Vi  ? 

15.  Arrange  in  order  of  magnitude,  Vli,  V6,  and  V175. 

16.  Arrange  in  order  of  magnitude,  V3,  Vl5,  and  ■\/25S. 

17.  Arrange  in  order  of  magnitude,  V3,  V5,  and  V7. 

MULTIPLICATION  OF  RADICALS. 
235.   1.  Multiply  V6  by  VW. 


V6  X  Vl5  =  V6  X  15  (§  226)  =V2x3x3x5 
=  V32  X  V2  X  5  =  3  VIO,  .4ns. 


208  ALGEBRA. 

2.  Multiply  V2^  by  </I^'. 

Reducing  to  equivalent  radicals  of  the  same  degree, 

V2^  X  v/4^^= (2  a)^  X  (4 a'^y  =  (2 a)^  x  (4 a2)t  =  V(2ay  x  v'(4a2)2    . 
=  y/2^a^  X  2*«*  =  ■v/26a6  x  \^a^  =  2  a  v^2  a,  Ans. 

3.  Multiply  V20  by  ^5. 

We  have,   V20  =  20^  =  20^  =  ^^203  =  \/(22  x  5)3  =  y/2^  x  58. 
Whence,  V20  x  v5  =  v/26  x  5^  x  5.=  v^26  x\/54  =  2x5^  =  2x5t 
=  2  x\/52  =  2v^,  ^ws. 

From  the  above  examples,  we  have  the  following  rule : 

Reduce  the  radicals,  if  necessary,  to  equivalent  radicals  of 
the  same  degree. 

Midtiply  together  the  expressions  under  the  radical  signs, 
a7id  write  the  result  under  the  common  radical  sign. 

The  result  should  be  reduced  to  its  simplest  form. 

EXAMPLES. 
Multiply  the  following : 

4.  V3  and  V48.  14.  ^9  and  -^135. 

5.  ^6^  and  ^36^.  15.  ^43  and  -^56. 

6.  Vii  and  VT8.  ^g  ^3^  ^^^  ^2f^ 

7.  VW  and  V50.  ^^  ^^  ^^^  ^3^ 

8.  V44  and  V27^.  ,^     3/^—       .    9/j^ 

18.    •voaa?  and  v4oic. 

9.  V3-0^6  and  VWTc.         ^^^   ^^  ^^^  ^_ 


10.  V36  and  V48. 

11.  VW  and  ^/63. 

12.  -v^  and  -^132. 


13.   VI  and  V^^ 


20.  ^4^  and   </W¥c. 

21.  a/45  and  -5/9. 

22.  V12  and  ^3. 


8  23.    V54  and  V72. 


RADICALS.  209 

24.  yjl  and  ^.  ^7.  V6,  ^,  and  ^. 

/^  /o  28.  V2,  ^/'i,  and  a/3. 

25.  X  —  ^^^  \/-- 

26.  Vab,  -i/bc,  and  A^ca.         """  ''  ^"^   ^9'  "         ^8 


..,-        ,    3,_         29.    ^,  ^,  and  #• 


30.   Multiply  2V3+3V2  by  3  V3  -  V2. 

2V3  +  3V2 
3V3-    y/2 


18  +  9V6 

-  2  V6  -  6 
18  +  7  V6  -  6  =  12  +  7  V6,  ^ns. 

Note.  It  should  be  remembered  that  to  multiply  a  radical  of 
the  second  degree  by  itself  simply  removes  the  radical  sign;  thus, 
\/3  X  V3  =  3. 


31.   Multiply  3V^+l+4a;  by  ^^x'  +  l-x. 


3V»2  +  1  +4aj 
2Vx2  +  1  -    a; 


6(x2  +  l)  +  8a;Vx2  +  l 

-3xVx2  +  1  _4a;2 


6x2  +  6     +5xVx2+ 1 -4x2  =  2x2  +  6 +  6x\^5nn,iin». 
Multiply  the  following : 

32.  5-2V3  and  4  +  3V3. 

33.  2V^+-3V2  and  Q^-^. 

34.  7 V2  -  4V5  and  3V2  -  8V5. 

35.  6Va  + 11  V'6  and  9Va-5V6. 

36.  5a/4  +  3a/5  and  6^/2  +  7^/25. 

37.  Va  +  2V6-3Vc  and  Va-2v^-3Vc. 


38.   4Va;  +  l  -6Vaj-l  and  3V^cl^ -2Va;- 1; 


210  ALGEBRA. 

39.  V6  -  V3  -  V5  and  V6  +  V3  +  V5. 

40.  9^1 -SV^  and  3^1 +  10 VI 

41.  5  V3  +  3  V5  +  4  V7  and  5  V3  +  3  V5  -  4  V7. 

42.  3 V2  -  4 V5  -  2V7  and  6 V2  -  8 V5  +  4 V7. 

43.  5  V8  +  6  Vi2  -  2  V20  and  7  V2  -  3  V3  +  4  V5. 

Expand  the  following  by  the  rules  of  §§  78,  79,  or  80: 

45.  (3V5  +  4)2.  48.    (7ViO  +  5V7)l 


46.    (5-2V3)2.  49.    (V2a+V3a-4)^ 


47.    (6V2-4V6)2.  60.    (S^x -\- y  -  2Vx^y. 

51.  (3V2  +  7)(3V2-7). 

52.  (6V3+4V5)(6V3-4V5). 

53.  (2V^rri  +  5V^)(2V^Tl-5V^). 


54.    ( Va  +  6  +  Va  —  6)  (Va  +  6  —  Va  -  6). 


55.    (3V2a  -  5  +  4 V4  a  -  3)  (3 V2  a  -  5-  4V4a  -  3). 

DIVISION  OF  RADICALS. 

236.  By  §  226,       V^  =  ^  x  V6. 

Whence,  ^=Vb. 

We  then  have  the  following  rule : 

Reduce  the  radicals,  if  necessary,  to  equivalent  radicals  of 
the  same  degree. 

Divide  the  expression  under  the  radical  sign  in  the  dividend 
by  the  expression  under  the  radical  sign  in  the  divisor,  and 
write  the  result  under  the  common  radical  sign. 


RADICALS. 


211 


EXAMPLES. 

1.  Divide  \/l5  by  V6. 

Reducing  to  equivalent  radicals  of  the  same  degree,  we  have 

V6       6^       6^       V(2  X  3)3      >'23x33     A'23x3~>24* 

2.  Divide  VlO  by  \/40. 

We  have,       vlO  =  10^  =  10^  =  VW  =  y/(2  x  6)8. 


-471S. 


Whence,   :^  =  ^?i2LM  =  ^  =  5I  =  5*  =  ^,  4,«. 


Divide  the  following : 

3.  V84  by  V7.        6. 

4.  Vl2  by  Vl5.      6. 
9.  ^8l  by  ^9. 

10.  -y/W^  by  ^39a3. 

11.  </l92  by  a/3. 

12.  V2  by  ^8. 


V56  by  V32.       7.    a/5  by  \/T3o. 

's/162  by  \/2.       8.    \/63  by  ^35. 

19. 


13.  </5ab  by  </125bh. 

14.  V28x  by  a/42^. 

16.  V2j|byV2J;. 

17.  -C/3^  by  </7^. 


6a^  by  V9a^a;\ 

22.  -^li^  by  ^'2S^\ 

23.  ^/20  by  -^125. 

OA         12/27    ,  4/3 

25.    ^/12^  by  V8^^ 


212  ALGEBRA. 

INVOLUTION  OF  RADICALS. 

237.   1.  Raise  Vl2  to  the  third  power. 

(v^)8  =  (12^)3=12^(§219)=12^=Vl2  =2\/3,  Ans.  . 

2.    Raise  -i/2  to  the  fourth  power. 

(^)4  =  (2^)4  =  2^  =  -k/¥  =  \/m,  Ans. 

We  then  have  the  following  rule : 

If  possible,  divide  the  index  of  the  radical  by  the  expoyient 
of  the  required  power;  otherwise,  raise  the  expression  under 
the  radical  sign  to  the  required  power. 

EXAMPLES. 
Find  the  values  of  the  following :  * 

3.  {-V^y.  7.   (2V^)'.  11.   (^128)3. 

4.  (a/5)2.  8.    (^7^)^  12.    (binVM^y. 

5.  (\/^+^)2.  9.    iVW^y.  13.    (V^a-2f. 


6.    (3V16)2.  10.    (V3)^.  14.    (V48aj3/)». 

EVOLUTION  OF  RADICALS. 

238.   1.  Extract  the  cube  root  of  -^27  ^. 

2.   Extract  the  fifth  root  of  a/6. 

^(  v^)  =  (6^^  =  6tV  =  '^^6,  Ans. 

We  then  have  the  following  rule : 

If  possible,  extract  the  required  root  of  the  expression  under 
the  radical  sign ;  otherwise,  multiply  the  index  of  the  radical 
by  the  index  of  the  required  root. 

Note.  If  the  radical  has  a  coefficient  which  is  not  a  perfect  power 
of  the  degree  denoted  by  the  index  of  the  required  root,  it  should  be 
introduced  under  the  radical  sign  (§  231)  before  applying  the  rule. 

Thus,  -s/^y/2  =  a/\/32  =  \/2. 


RADICALS.  213 

EXAMPLES. 
Find  the  values  of  the  following : 

3.  V(-v/49).  7.  V(16^/9).  ll.^-s/(2a</2a). 

4.  a/(V10).  8.  </(-</2B).  12.  ^/(27  a^^5^). 


6.    V(-V32a«).        9.  V(VV-6^+9).    13.  V(V125). 
W  6.    -yCv^Sl^).    10.  </{S^2^).  14.  ^(3a2</9^). 

TO  REDUCE  A  FRACTION  HAVING  AN  IRRATIONAL 
DENOMINATOR  TO  AN  EQUIVALENT  FRACTION  WHOSE 
DENOMINATOR  IS  RATIONAL. 

239.   Case  I.     When  the  denominator  is  a  monomial. 

The  reduction  may  be  effected  by  multiplying  both  terms 
t)f  the  fraction  by  a  radical  of  the  same  degree  as  the  de- 
nominator, having  under  its  radical  sign  an  expression 
which  will  make  the  denominator  of'  the  resulting  fraction 
rational. 

1.   Reduce   3  __^'  to  an  equivalent  fraction  having  a  ra- 


^3a^ 
tional  denominator. 

Multiplying  both  terms  of  the  fraction  by  \/9a,  we  have 

5  6\/9a  5\/9a      bWa 


\/§l2      </^^'9a      v/27a8        3a 


EXAMPLES. 


-,  Ans. 


Reduce  each  of  the  following  to  an  equivalent  fraction 
having  a  rational  denominator : 

2.  -^.  4.  -^.         6.       ^^     .       8.  A 


V6  V2b  </4.a^b  -v^ 

1      .       5      4^,        7    J_.  9  3        ^ 

V7a6«  .'   ^/3a:2  *   .J/27  *   -y/Us^y^ 


214  ALGEBRA. 

240.   Case  II.    When  the  denominator  is  a  binomial  con- 
taining  radicals  of  the  second  degree  only. 

1.   Reduce  ;=  to  an  equivalent  fraction  liavinsr  a 

5+V3 
rational  denominator. 

Multiplying  both  terms  of  the  fraction  by  5  —  Vo,  we  have 

5-V3_         (5_V3)2 

5  +  V3  ~  (5  +  \/3)  (5  -  \/3) 

^25-10V3  +  3  =28-10V3^14-5V3   ^^^^ 

25-3         ^^^      '       ^  22  11 


2.  Reduce  -^^ — to  an  equivalent  fraction  hav- 

2Vx  -  3 V^^^ 
ing  a  rational  denominator. 

Multiplying  both  terms  of  the  fraction  by  2y/x  -{■  SVx  —  I, 

Sy/x  -  2Vx~^  ^  (SVx  -  2Vx'^)(2Vx  +  3Vx^^) 
2 VS  -  3 Vx'^      (2 y/x  -  Zy/x^^) (2Vx-\-  3 Vx^H:) 

^  6  a;  +  b-\/xy/x^^  _  6(a;  -  1)  ^  6  +  by/^^^  ^^^ 
4x-9(a;-l)  9-bx      ' 

We  then  have  the  following  rule : 

Multiply  both  terms  of  the  fraction  by  the  denominator  with 
the  sign  between  its  terms  chariged. 

EXAMPLES. 

Reduce  each  of  the  following  to  an  equivalent  fraction 
having  a  rational  denominator : 

3.  ^   _.  6.   ^-^.  9. 
3+V5                Vx+A'^ 

5  y    5V2+V6  ^Q 


2V3-4  3V2-V6 

^    Va+^  g    3V5  +  2V2         ^^ 

'    Va-b  '   3V5-2V2*  '  -Vo^^-Va 


5V3 

+  4V5 

V*- 

-2  +  1 

Va;- 

-2  +  2 

V^ 

-6+Va 

RADICALS.  215 


12    2^+ Vl  —4:x\  jg     Vx  —  y  +Vx-\-y 

2a;  —  Vl  —  4:X^  ^x  —  y—  Va;  +  // 


13    ^t  —  Va-  —  V^^  ,g     Vcr  —  ar'  —  Va^  +  -t*^'. 


14    Vl  +  ct  —  Vl  —  g  j^w    4Va;  —  1  +  Va;  +  1 


Vl  H-  a  4- Vl  -  a  3 Va;  -  1  -  2Va;  +  1 

241.  The  approximate  value  of  a  fraction  whose  denomi- 
nator is  irrational  may  be  conveniently  found  by  reducing 
it  to  an  equivalent  fraction  with  a  rational  denominator. 

1.   Find  the  approximate  value  of to  three  places 

of  decimals.  2  —  V2 

1       _  2-f^  -2 +  v^-2  + 1.414^=  1.707  ...,^n.. 


2-V2      (2-V2)(2  +  v^)        4-2 

EXAMPLES. 

Find  the  approximate  value  of  each  of  the  following  to 
three  decimal  places : 

„       3  g    _2_  g    3V2-V6 


ViO  </25  3V2+V6 

3          1  g            5  g    3V3-f-2V5, 

V6-2  ■  V3-2V2  '   3V3-2V5 

4.    _t_.  7  V5  4-V2  jQ     V6-4V3 

3  +  2V5*  '  V5-V2*  *   2V6  +  5V3' 


PROPERTIES  OF  QUADRATIC  SURDS. 

242.  A  Quadratic  Surd  is  the  indicated  square  root  of  an 
imperfect  square ;  as  V3. 

243.  A  quadratic  surd  cannot  he  equal  to  the  sum  of  a 
rational  expression  and  a  quadratic  surd. 


216 

ALGEBRA. 

For,  if  possible,  let 

-Va  = 

■■b-{-Vc. 

Squaring  both  members, 

a  = 

:b'-{-2b\'c 

4-c. 

Transposing, 

2b-y/^  = 

a-b'-c. 

Dividing  by  2  b, 

Vc  = 

a  —  W  —  c 

O  7. 

We  then  have  a  quadratic  surd  equal  to  a  rational  ex 
pression,  which  is  impossible. 

Hence,.  Va  cannot  be  equal  to  6  -f-  Vc. 

244.   If  €i-\-  V6  =  c  -h  V^,  then  a  =  c,  and  V&  =  \^d. 

If  a  is  not  equal  to  c,  let  a  =  c  +  x. 

Substituting  this  value  in  the  given  equation,  we  have 

»  c-{-  x-{-  -y/b  =  c  +  Vd, 

or,  X  +  V6  =  Vd. 

But  this  is  impossible  by  §  243. 
Hence,  a  =  c,  and  therefore  Vb  —  V^. 


245.   7/*Va+Vfe=V^  +  Vy,  then  '^a  —  Vb  =  V^  —  -y/y. 
Squaring  both  members  of  the  given  equation,  we  have 

a  +  V&  =  a;  +  2^xy  +  y. 
Whence  by  §  244,       a  =  x-\-y,  (1) 

and  V6  =  2V^.  (2) 

Subtracting  (2)  from  (1), 

a  —  Vfe  =  a;  —  2  Va^  4- 2/- 
Extracting  the  square  root  of  both  members,  ^ 


Va  —  Vft  =  Va;  —  V^- 


oUe^Wi 


RADICALS.  217 

246.  Square  Root  of  a  Binomial  Surd. 

The  preceding  principles  may  be  used  to  find  the  square 
root  of  a  binomial  surd  whose  first  term  is  rational. 

Example.     Required  the  square  root  of  13  —  Vl60. 
Assume,  Vl3  -  Vl6U  =  Vx-Vy.  ( 1 ) 

Then  by  §  245,  Vl3+Vl60  =  Vx  +  y/y.  (2) 

Multiplying  (1)  by  (2),  Vl69  -  160  =  x-y. 

Or,  x-y  =  S.  (3) 

.  Squaring  (1),  IZ  -  VWO  =  x  -  2Vxy -\- y. 

Whence  by  §  244,  x  +  y  =  IS.  (4) 

Adding  (3)  and  (4),  2x  =  16,  or  x  =  8. 

Subtracting  (3)  from  (4),  2y  =  10,  or  y  =  5. 

Substituting  m  (1) ,       Vl3  -  \/l60  =  VS  -  V5  =  2  v^  -  VS,  Ans. 

247.  Examples  like  that  of  §  246  may  be  solved  by  in- 
spection by  putting  the  given  expression  into  the  form  of  a 
perfect  trinomial  square  (§  96),  as  follows : 

Reduce  the  surd  tei'm  so  that  its  coefficient  may  be  2. 

Separate  the  rational  term  into  two  parts  whose  product 
shall  he  the  expression  under  the  radical  sign  of  the  surd  term. 

Extract  the  square  root  of  each  part,  and  connect  the  results 
by  the  »ign  of  the  surd  term  (§  97). 

■  1.   Extract  the  square  root  of  8  4- V48. 


We  have,     Vs  +  V48  =  V8+2\/l2. 

We  then  separate  8  into  two  parts  whose  product  is  12. 
The  parts  are  6  and  2. 


Whence,      Vs  +  \/48  =  Vg  +  2\/12  +  2  =  Ve  +  \/2,  Ans. 

2.   Extract  the  square  root  of  22  -  3  V32. 

We  have,  V22  -  3  V32  =  V22  -  \/9  x  8  x  4  =  V22  -2V72. 
We  then  separate  22  into  two  parts  whose  product  is  72. 
The  parts  are  18  and  4. 

Whence,    V22  -  3\/32  =  VTs  -  Vi  =  3  V2  -  2,"  Ans. 


218  ALGEBRA. 

EXAMPLES. 
Extract  the  square  roots  of  the  following : 

3.  1H-2V28.          9.    12-VI08:  15.  56  +  5V48. 

4.  17-2V72.        10.   11+VI20.  16.  35-12V6. 

5.  49  +  2V48.        11.   26  +  2VI60.  17.  37-V640. 

6.  19  +  4V21.        12.   20-6V1I.  18.  35-20V3. 

7.  28-8V6.          13.   46-3V20.  19.  85  +  5V120. 

8.  30- 2  V56.        14.   35  +  lOViO.  20.  75-3V96. 

21.   2a;4-2V^^^=l.  •  22.    a-2Vab^^'. 

IMAGINARY  NUMBERS. 

248.  An  even  root  of  a  negative  number  is  impossible; 
for  no  number  when  raised  to  an  even  power  can  produce 
a  negative  result  (§  186). 

An  Imaginary  Number  is  an  indicated  even  root  of  a 
negative  number;  as  V—  4,  or  V— «"'^. 

In  contradistinction,  all  other  numbers,  rational  or  irra- 
tional, are  called  i-eal  numbers. 

249.  Every  imaginary  square  root  can  be  expressed  as 
the  product  of  a  real  number  by  V— 1. 

Thus,  V^^  =  Va'x(^l)  =  Va^  x  V^=^  =  a  V^l^; 


V-5  =V5  x(-l)  =  V5  xV-1;  etc. 

250.   To  find  the  positive  integral  powers  of  V—  1. 
By  §  189,  V—  1  signifies  an  expression  Avhose  square  is 
equal  to  —  1. 

That  is,  .  (V^'  =  -l. 


RADICALS.  219 

Then, 

(V'^>'=(V^)'x  V-1   =       1  xV^=V^;  etc. 

Thus,  the  first  four  positive  integral  X30wers  of  V  —  1  are 
V—  1,  —1,  —  V— 1,  and  1;  and  for  higher  powers  these 
terms  recur  in  the  same  order. 

251.  Addition  and  Subtraction  of  Imaginary  Numbers. 

Imaginary  numbers  may  be  added  or  subtracted  in  the 
same  manner  as  other  radicals.     (See  §  233.) 


1.    Add  V^4  and  V-36. 

V34  4-  ^/33B  =  2V^n.  +  6 V^^  (§  249)  =  8 V'^l,  Ans. 

EXAMPLES. 
Simplify  the  following : 


2.  V^Ii6+V^r25.  7.  V^+V^^^-V=^. 


3.  V^+V^^27.  8.  V^36-V=300  +  V-81. 


4.  V-18-V-8.  9.  V-a2-V-4a2-V^^9^ 


5.  V^ra2-V-(a-6)2.  10.  V-20-h V^-45- V-^. 


6.  V-aj2-f-V-y'4-V^2'.      11.  V-l-2a;-iB2-V-4^. 

252.  Multiplication  of  Imaginary  Numbers. 
The  product  of  two  or  more  imaginary  square  roots  may 
be  found  by  aid  of  the  principles  of  §  §  249  and  250. 

1.   Multiply  V^^  by  V^^. 

Vir2  X  V^^  =  V2  V^n[  X  VS  V^  (§  ii49) 

=  v^  V3(V^)2  =  V6  (-!)(§  260)  =  - V6,  Ans 


220 


ALGEBRA. 


2.  Multiply  together 
y/~9  X  V^n^  X  V^25  =  SV^n;  X  4\/^l  X  5y/^^ 

=  60(\/^=T)3  =  60(-  V^T)  (§  250) 

3.  Multiply  2  V^^  +  V^^  by  V^^  -  3  V"^. 

y/^2  -  3  V^=^ 

2(-2)  +  V2  V5(V^nr)2 

-  6\/2  V5  (  V^^)2 -3(- 5) 


4-5\/l0(-l) 


+  15  =  11  +  5\/l0,  ^ns. 


Note.     It  should  be  remembered  that  to  multiply  an  imaginary 
square  root  by  itself  simply  removes  the  radical  sign  ;  thus, 


V-2  x\/-2 


2. 


EXAMPLES. 
Multiply  the  following : 

4.    V~^^4^  by  V^^.  8.    - 


V^^by  V-12. 


5.  V- 9^2  by  -^-16ai     9.    -  V-72  by  -  V-50. 

6.  V^=^  by  V^=l0.  10.   2-5V^  by  3+4V^. 

7.  -V'^T^  by  -V^=~&.       11.   8+-V^by  7-5V^^. 

12.  4V^=^  -  7 V^^  by  2V^=^  -  V^^. 

13.  2  V^^  +  3  V^^  by  4  V^^  -  6  V^=^. 


14.    V— aj^,  V— 2/^  and  V—z^. 
15. 


_8,   _V-18,  and  V-32. 

"3+V^^. 


16.    V-  10  +  5V-  5  by  3 
17. 


2+V-3  by  V^r8-V-12. 


RADICALS. 


221 


18.  V-1,  V^36,  V-04,  and  V- 100. 

19.  V^"2,  V^   -V"^  and  V^^. 

Expand  the  following  by  the  rules  of  §§  78,  79,  or  80: 

20.  (H-V'=^)2.  23.    (2^/^^-SV^^y. 

21.  (5V^^  +  2V^'.      24.    (a;+V^)(a;-V^). 

22.  (4-V^l  25.    (5  4-6V^n:)(5-6V^). 
26.    (3V^^  +  2V^(3V^^t-2V^. 

27..  (7V^=^  +  4V^^)(7V^^-4V^. 

^    28.  (V:r2  +  ^v^-+(V^=^-2V'^)^ 

Reduce  each  of  the  following  to  an  equivalent  fraction 
having  a  rational  denominator : 

5  V^^  +  4  V^^ 


29. 


30. 


1+V^^ 
1-V^ 


31. 
32. 


5V-2-4V^^ 
3V^=^  -h  2 V^Ts 


Expand  the  following  by  the  rule  of  §  188 : 
33.   (1-hV'^^)^  34.    (2-V^^ 

253.  Division  of  Imaginary  Numbers. 
1.   Divide  V^^^^  by  V^^. 

We  have,  ^^31  =  VT2x^  ^  Vl2  ^  ^  ^  2,  Ans. 

v^=r3     vs  V- 1     V3 


2.   Divide  VlO  by  V-  2.' 

Vio  _  -  VTo (-!)_-  VIo  ( V^  )-^ 


V-2        V2\/^ 


\/2V-l 


250)  =  ->/5>/-l; 


222  ALGEBRA. 

EXAMPLES. 
Divide  the  following : 

3.  V^^^25  by  V^=^.  8.  -Va  by  V^=^l 

4.  V^=r32  by  V^^.  9.  </^^75  by  ■</'^^. 

5.  V42  by  V^^.  10.  -</^^lS  by  "v^^^. 

6.  V63  by  -V^=T.  11.  -^3108  by  ^/^^. 


7.    V-a&byV-&c.        12.    -V-40by-V-o. 
SOLUTION  OF  EQUATIONS   CONTAINING  RADICALS. 


254.   1.    Solve  tlie  equation  Var^  —  5  —  x  =  —  l. 


Transposing  —  x,  Vx^  —  5  =  x  —  ,1. 

Squaring  both  members,         x^  —  5  =  x^  ~  2  x  -{-  1. 
Transposing  and  uniting  terms,  2x  =  6. 
Whence,  ic  =  3,  Ans. 


2.    Solve  the  equation  v2  a;  +  14  +  V2  a;  +  35  =  7. 


Transposing  y/2x  +  14,    V2F-f36  =  7  -  \/2  a;  +  14. 


Squaring  both  members,      2  x  +  35  =  49  -  14v  2  x  +  14  +  2  x  +  14. 
Transposing  and  uniting  terms, 


14\/2x+14  =  28. 


Or,  ,         \/2  X  +  14  =  2. 

Squaring  both  members,      2  x  +  14  =  4. 

2x  =  -10. 
Whence,  x  =  ~  5,  Ans. 

From  the  above  examples,  we  derive  the  following  rule : 
Transpose  the  terms  of  the  equation  so  that  a  radical  term 

may  stand  alone  in  one  member ;  then  raise  both  members  to  a 

power  of  the  same  degree  as  the  radical. 

Tf  radical  terms  still  remain,  repeat  the  operation. 

Note.     The  equation  should  be  simplified  as  much  as   possible 
before  performing  the  involution. 


^^  ^  RADICALS.  223 

EXAMPLES. 
Solve  the  following : 


3.    V3a;-5-2  =  0.  7.    V^  +  4  4-Va;  =  3. 


5.  V9x2^5_33.^1  9    V5a;  4- 10 -V5lc  =  2. 

6.  V«  -  Vx  - 12  =  2.  10.    Va;H-ll+V^T6  =  5. 


11.   -^— -^— =V63^. 

V3  -  a;      V6  -  a; 

12.  V2a;-l+V2a;  +  4  =  5.   15.  V^^+5'-\-Vx^^=2Vx. 

13.  V^  +  V^T4  =  A.  16.  2V.-3^4V..-4. 

Vx  3Vx  +  2     6v'a;  + 1 

14.  V^36+Vi  =  _J=.    17.  V3^^+l+V3^^4. 

Va;  —  6  V3  a;  +  1  —  V3^ 


18.    Var'-5a;-2+Va^  +  3a;  +  6  =  4. 
19.  V^_V^^  =  -i^.       21.  ^^^+^-^^ 


Va?  —  5  Va;  +  a  +  Va;  —  a 


20.  Va;-hl5-Va;H-3  =  2Va;.     22.   Va;  +  a  + Va;-a  =  26. 


I.    Vaj  +  2  a  —  Va;  —  3  a  =■ 


-y/x  —  Sa 


24.  V;r  +  6  -h  Va;  +  9  =  V4a;  4-  29. 

25.  Va;  +  aH-Va;  +  4a  =  2 Va;  +  2 a. 


26.    Vex  4-  a&  +  Vca;  —  ah  =  V4  ca;  —  2  a6. 


^  27.    Va;  +  V  a  -  Va  a;  H- ar^  =  Va. 
28.    V(a;  +  2aV4x+Ta')  =  V^-2a. 


29.   2  (a?  4-  a)  (^-  4-  Va;'  -  a')  =  a^ 


^     30.    Va;  4- 1  4-  Va;  4-  5  =  Va^  +  2  4-  Va;  4-  3. 
31.    Vfi  —  a;  +  V6  —  a;  =  Va  4-36  —  43;. 


224  ALGEBRA. 


XXII.   QUADRATIC  EQUATIONS. 

255.  A  Quadratic  Equation  is  an  equation  of  the  second 
degree  (§  158)  containing  but  one  unknown  quantity. 

A  Pure  Quadratic  Equation  is  a  quadratic  equation  in- 
volving  only   the   square   of    the   unknown   quantity ;    as 

An  Affected  Quadratic  Equation  is  a  quadratic  equation 
involving  both  the  square  and  the  first  power  of  the  un- 
known quantity ;  as  2x^  —  3x  —  5  =  0. 


PURE  QUADRATIC  EQUATIONS. 

256.  A  pure  quadratic  equation  may  be  solved  by  reduc- 
ing it,  if  necessary,  to  the  form  of  =  a,  and  then  extracting 
the  square  root  of  both  members. 

1 .  Solve  the  equation  3  x^ -}- 7  =  — -\- 35. 

Clearing  of  fractions,  12  ic2  +  28  =  5  a;2  +  140. 

Transposing  and  uniting  terms,  7  aj^  =  112. 

Or,  x2  z=  16. 

Extracting  the  square  root  of  both  members,  we  have 

X  =  ±  4,  Ans. 

Note  1.  The  sign  ±  is  placed  before  the  result,  because  the  square 
root  of  a  number  is  either  positive  or  negative  (§  192). 

2.  Solve  the  equation  7  ic^  —  5  =  5  a;^  —  13. 

Transposing  and  uniting  terms,  2  ic^  =  —  8. 

Or,  a;2  =  -  4. 

Whence,  x  —  ±\/—  4  =±  2V'—  1,  Ans. 

Note  2.  In  Ex.  2  the  values  of  x  are  iinaginary  (§  248)  ;  it  is 
impossible  to  find  any  real  values  of  x  which  will  satisfy  the  given 
equation. 


QUADRATIC   EQUATIONS.  225 

EXAMPLES. 

Solve  the  following  equations : 

3.   3x'-26  =  9x^-S0.  5.   3(x -^1)  -  x(x -1)  =  4:X. 

4     X_Ji  =  ^        '    "^  6     ^4-?  =  -^4-l? 

'   Sx"     dx"     S  '   3     X     12^  x' 

7.  (2a;-3)(a.'-f7)  + (2fl;  +  3)(a;-7)  =  58. 

8.  5  (x  -h  a)  (a;  —  ft)  +  4  ax  =(x-\-2  af. 

9.  (3a;  +  2)  (4a;  -  5)  -  (5a;  +  3)(6a;  -  5)  +  45  =  0. 
irt    2a;2_^4     3a^_7      n  /  ^  .     . 


5  3  15  Va-  +  aj' 


11.    Vl0  +  a;-Vl0-a;  =  2.   13.    V2a;  + 8 +  2\^-h5  =  2. 

+  a;  +  l     a;^-a;-i- 
a;  —  1  a;-|-  1 

l±3_ 

10  25 


j^    ar»  +  a;  +  l     ^-^  +  l^g 


^      15    3ar^-2     5a.-^  +  3     4ar^-4^Q 


16    ?^zi?^3^_6^±i 
6  2       9ar'-f7* 

17    a? +  4     a; -4^  10  ^g     a;* -3ar^  +  4  _  a;2_  3 


a;_4      a;  +  4      3  3a;*  +  2a;2_,4     3a^^2 


19.   a; Var*  +  12  +  x^x^  +  6  =  3. 
a;  —  6     a;-f6  o?  —  h^ 


21.    Vl+a;4-ar'-Vl-a;  +  ar^  =  V6. 

22    ^'"^  ^  _i_  ^'  ~  Q^  _  ti  +  &  ,  a  —  h 
X  —  a      x  +  a     a  —  b     a  +  b 

(First  add  the  fractioift  in  the  first  member ;  then  the  fractions  in 
the  second  member.) 


226  ALGEBRA. 


AFFECTED   QUADRATIC  EQUATIONS. 

257.  An  affected  quadratic  equation  may  be  solved  by 
reducing  it,  if  necessary,  to  the  form  .^•^  -\-px  =  q. 

We  then  add  to  both  members  such  an  expression  as  will 
make  the  first  member  a  perfect  trinomial  square  (§  96)  ; 
an  operation  which  is  Mimed  completing  the  square. 

258.  First  Method  of  Completing  the  Square. 

Example.     Solve  the  equation  ic^  +  3  a^  =  4. 

A  trinomial  is  a  perfect  square  when  its  first  and  third 
terms  are  perfect  squares  and  positive,  and  the  second 
term  plus  (or  minus)  twice  the  product  of  their  square 
roots  (§  96). 

Then,  the  square  root  of  the  third  term  is  equal  to  the 
second  term  divided  by  twice  the  square  root  of  the  first. 

Hence,  the  square  root  of  the  expression  which  must  be 

added  to  a^  +  3  .t  to  make  it  a  perfect  square  is  -— ,  or  -• 
Adding  to  both  members  the  square  of  -,  we  have 

Extracting  the  square  root  of  both  members  (§  97), 

x^\  =  ±\'     (See  Note  1,  §  256.) 

Transposing  -,  ^  =  ~  2  "^  2'  2  ~  2* 

Whence,  a;  =  1  or  —  4,  Ans. 

From  the  above  example,  we  derive  the  following  rule  : 
Reduce  the  equation  to  the  form  x^  -\-px=  q. 
Complete  the  square  by  adding  to  both  members  the  square 
oj  half  the  coefficient  of  x. 

Extract  the  square  root  of  both  iixiembers,  and  solve  the  sim- 
ple equations  thus  formed. 


QUADRATIC   EQUATIONS.  227 


259.  ,  1.    Solve  the  equation  3  a;-  —  8  a;  =  —  4. 


Dividing  by  3,  x^-^  =  --;  ^^X^'^^  ' 

3  3,  -<I 

which  is  in  the  form  x-  -\-  px  ■=  q. 

Adding  to  both  members  the  square  of  |,  we  have 

2_8£     /4\2__4      16_£ 
"^        3  ^Uj    "     3^9"-9 

4  2  ^ 

Extracting  the  square  root,  x  —  =  ±  -• 

o  o 

4      2  2 

Whence,  x  =  -±-  =  2or-,  Ans, 

o       o  o 

If  tlie  coefficient  of  a-^  is  negative,  the  sign  of  each  term 
must  be  changed. 

2.   Solve  the  equation  -  9  ar'  -  21  x  =  10. 

Dividing  by  -  9,  x2  + 1^  =  -  1^. 

Adding  to  both  members  the  square  of  |,  we  have 

\q)   ^      9    '  36  "36 


^.  +  7x+f7y^_|^49^9 
7      .  3 


Extracting  the  square  root,    a;  +  -  =  ±  -• 

Whence,  x  =c-I  ±§  =  -?  or -^,  Ans. 

6     6         3  3 


EXAMPLES. 
Solve  the  following  equations : 

—  3.   x^  +  &x  =  l.  10.  2a^  +  ll.'c  =  -5. 

4.  a;2-4a7  =  32.  -11.  2a;2  +  9a;- 5  =  0. 

5.  a;2  +  lla;  =  -18.  12.  5a^H-8  =  22a;. 

6.  a.'2-13a;  =  -30.  13.  20  -  27  a;  =- 9  ar^. 

7.  ar^  +  aJ  =  30.  ^14.  7  -  10  a;- 8aj2  =  0. 

-  8.   3a;2_7^^_2.  15.  12 +  16.a;  -  3a;2^  q 

9.   4.x'-^x==l.  .16.  Q>x'  +  ^  =  -XXx, 


228  .  ALGEBRA. 

260.  If  the  coefficient  of  o?  is  a  perfect  square,  it  is  con- 
venient to  complete  the  square  directly  by  the  principle 
stated  in  §  258;  that  is,  by  adding  to  both  members  the  square 
of  the  quotient  obtained  by  dividiyig  the  coefficient  of  x  by  twice 
the  square  root  of  the  coefficient  of  x^. 

1.  Solve  the  equation  9  a;^  —  5  a:  =  4. 

Dividing  5  by  twice  the  square  root  of  9,  the  quotient  is  f . 
Adding  to  both  members  the  square  of  f ,  we  have 

\Ql  36      36 

5  IS 

Extracting  the  square  root,     3  a:  —  =  ±  — 

Transposing,  3x  =  -  ±  —  =  3  or  -  -• 

6      6  3 

4 
Whence,  a;  =  1  or  — ,  Ans. 

9 

If  the  coefficient  of  a^  is  not  a  perfect  square,  it  may  be 
made  so  by  multiplication. 

2.  Solve  the  equation  8  a^  —  15  a;  =  2. 

Multiplying  each  term  by  2,  16  x^  —  30  a:  =  4. 

Dividing  30  by  twice  the  square  root  of  16,  the  quotient  is  -3/,  or  J^^.. 

Adding  to  both  members  the  square  of  Y-,  we  have 


I  4  /  16       16 

A/ 

4'~^T       y  i/ 


Extracting  the  square  root,  4  x =  ±  —  '^ 


Transposing,  4x  =  — ±  —  =  8or 

4       4  2 

Whence,  x  =  2  or  — ,  Ans. 


Note.     If  the  coefficient  of  x^  is  negative,  the  sign  of  each  term 
must  be  changed. 


QUADRATIC   EQUATIONS.  229 

EXAMPLES. 

Solve  the  following  equations : 

3.  4:x'  +  7x  =  2.  10.  49af'-7a-  =  12. 

.4.  Wx'-\-S2x  =  -15.  ^11.  25a.-2-f  25.^•^-6  =  0. 

5.  9i«^-lla;  =  -2.  12.  120^  +  8a;  =  - 1. 

6.  Sx'-j-2x  =  3.  13.  32ic2^i^_12a... 

7.  5x'-{-16x  =  -3.  14.  28  +  5a;-3iB2^0 

8.  36ar^-36cc  =  -5.  15.  x'  +  l  =  20ar'. 

9.  64a;2^48^.^7  jg  4  +  3a;- 27a.'2  =  0. 

261.  Second  Method  of  Completing  the  Square. 
Every  affected  quadratic  can  be  reduced  to  the  form 

aa^  -\-bx  =  c. 
Multiplying  both  members  by  4  a,  we  have 

4  a  V  4-  4  abx  =  4  ac. 
Completing  the  square  by  adding  to  both  members  the 
squai-e  of  - — ^^  (§  260),  or  b,  we  obtain 

4aV  +  4  abx  +  b^=^b^  +  4ac. 
Extracting  the  square  root, 

2  oa;  +  6  =  ±  VfeM^Toc. 


Transposing,  2ax  =  —  b±  V6^  +  4  ac. 


Whence,  ^  =  :^^±V6^  +  4 


2  a  , 

From  the  above  example,  we  derive  the  following  rule :  Y^ 

Reduce  the  equation  to  the  form  ax^  +  bx  =  c. 

Multiply  both  members  by  four  times  the  coefficient  of  a^^ 
and  add  to  each  the  square  of  the  coefficient  of  x  in  the  given 
equation. 


230  ALGEBRA. 

Extract  the  square  root  of  both  members,  and  solve  the 
simple  equation  thus  formed. 

The  advantage  of  this  method  over  the  preceding  is  in 
avoiding  fractions  in  completing  the  square. 

262.   1.    Solve  the  equation  2ic^  —  7  a;  =  —  3. 

Multiplying  both  members  by  4  x  2,  or  8, 
16x2-56x=-24. 
Adding  to  both  members  the  square  of  7,  we  have 

16a;2  _  56x  +  72  =  -  24  +  49  =  25. 
Extracting  the  square  root,  4  a;  —  7  =  ±  5. 

4x  =  7±5  =  12  or  2. 

Whence,  x  =  3  or  ^,  Ans. 

If  the  coefficient  of  x  in  the  given  equation  is  even,  frac- 
tions may  be  avoided,  and  the  rule  modified,  as  follows  :     ■ 

Multiply  both  members  by  the  coefficient  of  x^,  and  add  to 
each  the  square  of  half  the  coefficient  ofx  in  the  given  equation. 

2.  Solve  the  equation  15  aj^  +  28  x  =  32. 

Multiplying  both  members  by  15, 

15%2+ 15(28  x)=  480. 

Adding  to  both  members  the  square  of  14,  we  have 

152x2  +  15(28  x)  +  142  =  480  +  196  =  676. 

Extracting  the  square  root,  15  x  +  14  =  ±  26. 

15x=-14±26=:12  or  -40. 

4  8 

Whence,  x  =  -  or  - -,  Ans. 

0  6 

EXAMPLES. 
Solve  the  following  equations  : 

3.  0^-7  a;  =  30.  6.   8a;2  +  14a;  =  -3. 

4.  2a^-f-5a;  =  18.  7.   10a^+7a;  =  -l. 

5.  3aj2-2a;  =  33.  8.    ^x'-2x  =  12. 


QUADRATIC   EQUATIONS.  231 

9.  4a^-7x  =  -3.  14.  6a^4-17ic  =  - 10. 

10.  6a^-llaj=10.  15.  5 x' -\- 15  =  2S x. 

11.  4a^-f  24aj  +  35  =  0.  16.  9 a.-^  =  32 a;  -  15. 

12.  4:x  +  4.  =  15x'.  17.  3-5a;-12x2  =  0. 

13.  4-15a;-4ar^  =  0.  18.  9ar^ -f  15a;  +  4  =  0. 

MISCELLANEOUS   EXAMPLES. 

263.  The  following  equations  may  be  solved  by  either  of 
the  preceding  methods,  preference  being  given  to  the  one 
best  adapted  to  the  example  under  consideration. 

3      1        13  ^     1 
Sx"     24a;         2* 

2.    ^-A  =  — ^.  4.    ?  +  ^=-??. 

4^a;  10 

5.  (3  a.' -f- 2)  (2  a;  4- 3)  =  (a; -3)  (2  a; -4). 

6.  9(aj-l)2-4(a;-2)2  =  44. 

7.  4(a5-l)(2aj-l)  +  4(2a:-l>(3a;-l) 

+  4(3x-l)(4a;-l)  =  53ajl 

8  ?2__M_  =  i  —10    a;  +  2     4-a;^7 

'     X      x-\-l        '  .     '    x-l        2x       3* 

9  _3 ?_=i  11      a;  +  l       2a;  +  1^17 

a;-6     a;-5        '  '    2a;-|-l      3a;  +  l      12* 

12.    (2aj  +  l)2-(3.'c-2)2-(a;-f  1)^  =  0. 

13.    V6  +  10a;-3ar^=2a;-3.    17     a;        x" -  6    ^6 

00^  11  3      3(a;  +  4)      x 

14    2  — 3a;     4  — a;_ll  ^  ^ 

4  a;_2~4  18.    Va; 4-2-1- V3a;  +  4  =  8. 

15.    (.-3)3- (.-,2)3= -65.      ^^     vi2^  3 


x' 
3 

X 

2" 

35 
6' 

1 
2' 

5 
6.2 

=  - 

7 
12a; 

16.    Va;-lH-V3a;-|-3  =  4.  5  2-hVl2-a; 


232  ALGEBRA. 


*^      21         1      I  ^   '^  I ~_^ =::0 

■  3a.'4-l      (3a;  +  l)(7a;-f  1) 

22        1  4^2 

•    a;_6      3(a?-l)  3a; 

4i»-3         2»     ~    * 

2^    a;  +  2  ,  a;  -  2  ^  6  a; +16 

a;_2'^a;  +  2  3a; 

12 


26.    V5  +  a;+V5- 


V5  —  a; 


26.  Va;  +  3-Va;  +  8  =  -5Va;. 

27.  ^.+     1  1  ' 


28.    3 


l_a;2'l  +  a;      1-a;  8 

13  5 


a;-h2      2(2  a; -3)      (a;  +  2)  (2  a;  -  3) 


29     ^^  ~  ^  _ ^  ~4a; _  o 

7  —  X       2a;  +  1~ 

3a;-6      7       ll-2a; 


30. 


5 -a;       2      2(5 -2a;) 


gj     3-2a;      2  +  3a;^l      16a;  +  ay^ 
2  + a;        2 -a;       3     .  a;^  -  4   ' 

32    ^  +  ^  _L  ^  +  ^  ^  2  a; +  13 
a;  —  1      a;  —  2         a;  +  l 

33.    V2^+9^T9+V2^T7^T5  =  V2. 


-    35 


34.    V3x  +  1  -  V4a;  +  5  +  Va;  -  4  =  0. 
1  15  a; 


(3a;+l)(l-5a;)     2(1  -  5  a;)  (7  a; +1)     (3a;+l)(7a;+l) 


36        Va;     _  Va;  +  2  _  5 
Va;  +  2         Vi         ^ 


QUADRATIC   EQUATIONS.  233 


37.    Vo  -  2X  +  V15  -  Sx  =^/26  -  5x. 

x^8      x-1^    x-2 


264.  Solution  of  Literal  Quadratic  Equations. 

For  the  solution  of  literal  affected  quadratic  equations, 
the  methods  of  §  262  will  be  found  in  general  the  most 
convenient. 

1.  Solve  the  equation  x^  -\-  ax  —  bx  —  ab  =  0. 
The  equation  may  be  written 

rc2  -f-  (a  -  b)x  =  ah. 

Multiplying  both  members  by  4  times  the  coeflBcient  of  x^, 

4ic2  +  4(a  -  b)x  =  Aah. 

Adding  to  both  members  the  square  of  a  —  ft, 

4 a;2  +  4 (rt  -  6)x  +  (a  -  6)2  =  4  a6  +  a2  -  2a6  +  62 

=  a2  +  2  a6  +  62. 
Extracting  the  square  root, 

2x4-(a-6)  =  ±(a  +  6). 

2«=-(a-6)±(a  +  6). 

Therefore,  2x  =  -a  +  6  +  a  +  6  =  2&, 

or  2x  =  -af6-a-6  =  -2a. 

Whence,  x  =  6  or  -a,  Ans. 

Note.  If  several  terms  contain  the  same  power  of  x,  the  coeffi- 
cient of  that  power  should  be  enclosed  in  a  parenthesis,  as  shown 
in  Ex.  1. 

2.  Solve  the  equation  {m  —  V)y?  —  2  m^x  =  —4:m^. 
Multiplying  both  members  by  m  —  1, 

(m  -  1)2x2  -  2  m^{m  -  l)x  =  -4  m2(w  -  1). 
Adding  to  both  members  the  square  of  m^, 

(m  -  1)2x2  r-  2  m2(m  -  l)x  +  m*  =  m*  -  4  m^  +  4  m^. 


234  ALGEBRA. 

Extracting  the  square  root, 

(m  —  \)x  —  mP-  =  ±  {m-  —  2  m). 

{m  —  l)x  =  m2  +  m^  --  2  m  or  m^  —  m^  +  2  »7i 
=  2  77i(m  —  1)  or  2  m. 

Whence,  a:  =  2  m  or  — ^,   ^?is. 

m  —  1 

EXAMPLES. 

Solve  the  following  equations  : 

3.  aj2-4aa;  =  962_4a2.  6.    x^ -\- ax -{- bx  +  ah  =  i). 

4.  a;^  +  2  mx  =  2  m  +  1.  7.   a^  —  m^ic  —  m%*  =  —  m'\ 

5.  x^  —  («  —  l)aj  =  a.  8.    aco;^  +  hex  —  ada;  =  6d. 

9.   x'-2ax-12x  =  Sa^-\^a-^^. 

10.  (a-5)a;2-(a  +  6>'  =  ~26. 

11.  (a  -  x)(a'  +  6^  _^  ax)  =  a^  +  6a^. 

12.     l+g  ^l-g  ^-^  13^    _^_4._fL_=2. 

1  —  cia;      1  -\-  ax  x  —  a     x  —  b 

14.  (aj4-2a)«-(a.'-3a)^  =  65a^ 

15.  (l-a^(x-{-a)-2a(l-xF)=0. 


16.  V(a-26)x  +  8a6  =  a;H-46. 

17.  6a^-(5ct  +  5)aj  =  -a2_a6  +  262. 


18.    Vx-{-7a-{-Vx  —  2a=-V5x-\-2a. 
-Q     ic  —  aaj  +  a     5aa;  —  3a  —  2_ 

IS.       ; 1 5 -g — U. 

.T-f-a     ic  —  a  a^  —  ar 

20.    Vx-12a6  =  ^^'~^'. 

21     ^!_±1  ^  2(aMi&!)^        22.^±^  +  ^-^  =  -. 
a?  o?  —  b^   '  '    x-\-b     x-\-  a     2 


23.    V2a;-4a  +  V5ajH-3a 


3a; 


V2  a;  —  4  a 


QUADRATIC  EQUATIONS.  235 

24.    x^-(a-b)x  =  (a-c){b-c). 


25.  V3 a;  +  2 a  —  V4 ic  —  6 rt  =  v2a. 

26.  f^^Y-7f^^^Vl2  =  0. 


rtiy     a;  4-V12a  — a;  _  Va  +  1 
X  —  Vl27t  —  a;      Va  —  1 

28.  (a-^/)aj2  +  (6-c)a;  +  (c-a)=0. 

29.  (a*-l)x'-2(a*-\-l)x  =  -a*-{-l. 

30.  eiJLl  =  .^Jzi  +  _^. 


31.  -l-  +  _i-  =  l  +  l. 

x— a      a;  — />      a      b 

32.  (c  +  a  -  2  6)  ar2  -f  (a  +  ^  -  2  c)  a;  +  /^  +  c  -  2  «  =  0. 

265.  Solution  of  Quadratic  Equations  by  a  Formula. 

It  was  shown  in  §  201  that,  if  a^  +  bx  =  c,  then         ^  ' 

^^-6±V6^4-4ac      c^  y     J      (^) 

This  result  may  be  used  as  a  formula  for  the  solution  of 
any  quadratic  equation  in  the  form  oar^  +  bx  =  c. 

1.  Solve  the  equation  2ar^  +  5a;  =  18. 

In  this  case,  a  =  2,  6  =  5,  and  c  =  18  ;  substituting  in  (1), 

5±V25Tig^-5±Vl69^-5i:l3^^  ^^   _9    ^^^^^ 
4  4  4  2 

2.  Solve  the  equation  110  a?^  -  21  a^  =  -  1. 

In  this  case,  a  =  110,  6  =  -  21,  and  c  =  -  1 ;  substituting  in  (1), 


^^21±V441-440^2_^±_1^^  ^^  1    ^^^ 
220  220        10        11 

Note.     Particular  attention  must  be  paid  to  the  signs  of  the  coeffi- 
cients ill  making  the  substitution. 


236 


ALGEBRA. 


EXAMPLES. 
Solve  the  following  equations :. 


3.  2x^-{-x=G. 

4.  x'-5x  =  36. 

5.  a^-\-Ux-{-4:S  =  i). 

6.  5x'-13x  =  -0. 

7.  6x''-x  =  5. 

9.  4:X^-21x  =  -27. 


10.  280^2 -hl6.^'  =  -l. 

11.  Sa^-\-4:lx-\-5  =  0. 

12.  16a-24- 16.1'- 5  =  0. 

13.  30a?-8  =  25a;2. 

14.  12x^-^7  =  -25x. 

15.  2-3a;-54.T2=:0. 

16.  3  +  14a;-24a^  =  0. 


266.  Solution  of  Equations  by  Factoring. 

Let  it  be  required  to  solve  the  equation 

(x-3)(2x-^5)=0. 

It  is  evident  that  the  equation  will  be  satislied  when  x 
has  such  a  value  that  one  of  the  factors  of  the  first  member 
is  equal  to  zero ;  for  if  any  factor  of  a  product  is  equal  to 
zero,  the  product  is  equal  to  zero. 

Hence,  the  equation  will  be  satisfied  when  x  has  such 
a  value  that  either 

a;  -  3  =  0,  (1) 

or  2x-\-5  =  0.  (2) 

5 
Solving  (1)  and  (2),  we  have  x  =  S  or 

It  will  be  observed  that  the  roots  are  obtained  by  placing 
the  factors  of  the  first  member  separately  equal  to  zero,  and 
solving  the  resulting  equations. 

267.  1.  Solve  the  equation  y?—Bx  —  2i  =  0. 

Factoring  the  first  member,   (a:  -  8)  (cc  +  3)  =  0.  (§  100) 

Placing  the  factors  separately  equal  to  zero  (§  266),  we  have 

X  -  8  =  0,  and  x  +  3  =  0. 
Whence,  a;  =  8  or  -  3,  Ans.  • 


QUADRATIC   EQUATIONS.  237 

2.  Solve  the  equation  2  a^  —  a;  =  0. 
Factoring  the  first  member,  x(2x  -  1)  =  0. 
Placing  the  factors  separately  equal  to  zero, 

X  =  0,  and  2  X  -  1  =  0. 

Whence,  x  =  0  or  -,  Ans. 

3.  Solve  the  equation  y?  -\- \q^  —  x  — \  =  ^. 

Factoring  the  first  member,  (x  +  4)  (x^  -  1 )  =  0.  (§  93) 

Therefore,  x  +  4  =  0,  and  x'^  —  1  =  0. 

Whence,  x  =  —  4  or  ±1,  Ans. 

4.  Solve  the  equation  or^  —  1  =  0. 

Factoring  the  first  member,  (x  -  1)  (x^  +  x  +  1)  =  0.  (§  103) 

Therefore,  x  -  1  =  0,  and  x^  +  x  +  1  =  0. 

Solving  the  equation  x  —  1  =  0,  we  have  x  =  1. 
Solving  the  equation  x^  -f  x  +  1  =  0,  we  have 

EXAMPLES. 
Solve  the  following  equations  : 

5.  a.'2  +  3a;-28  =  0.  10.  3 a.-^  +  24 .r^  =  0. 

6.  aj2_i4a;-|.45  =  0.  11.  16a^-9a;=0. 

7.  .T2_|_lia;^24  =  0.  12.  (2a;  +  5)(9ar^-49)=0. 

8.  a;2-6a;-72  =  0.  13.  \2^ -1  :^ -\^x  =  f). 

9.  5a;2-7a;  =  0.  14.  (.^•2- 8)(.'c24-4)=  0. 

16.   {x  -  3)(2  ^  -f- 13  a;  +  20)  =  0. 

16.  {x  -  3)(a;  +  4)(a;  _  5)  -  60  =  0. 

17.  (ar»-9a2)(2a;2  +  aa;-a2)=0. 


238  ALGEBRA. 

18.  x«  +  l  =  0.  22.   8a^  +  125  =  0. 

19.  a^-27  =  0.  23.   x«  -  64  =  0. 

20.  16 aj^- 81  =  0.  24.   ^xf  -  a^ -\-x -1  =  0. 


21.   27a^-64a3  =  0.  25.    ^x"  -  ^2x -{-1  =  x -1. 

26.  5a^-a;2- 125a; +  25  =  0. 

27.  8x3_^20a;2-18a;-45  =  0. 

28.  4:a^+5x'-\-72x-{-90  =  0. 


29.    Va  -\-x  -\-  V<x  —  ic  =  Va;. 


30.    Va  +  Va;  -  V  a  -  Va;  =  Va;. 

Note.  The  above  examples  are  illustrations  of  the  important  prin- 
ciple that  the  degree  of  an  equation  indicates  the  number  of  its  roots  ; 
thus,  an  equation  of  the  third  degree  has  three  roots ;  of  the  fourth 
degree,  four  roots  ;  etc. 

It  should  be  observed  that  the  roots  are  not  necessarily  unequal ; 
thus,  the  equation  x'-^—  2 cc  +  1  =  0  may  be  written  (x  —  1) (x  —  1)  =  0, 
and  therefore  its  two  roots  are  1  and  1. 

PROBLEMS. 

268.  1.  A  man  sold  a  watch  for  ^21,  and  lost  as  many 
per  cent  as  the  watch  cost  dollars.     What  was  the  cost  ? 

Let  X  =  the  number  of  dollars  the  watch  cost. 

Then,  x  =  the  per  cent  of  loss, 

and  X  X  -^ ,  or  ^—  =  the  number  of  dollars  lost. 

100         100 

X'2 

By  the  conditions,  —  =  x  —  21. 
^  100, 

Solving,  X  =  30  or  70. 

Then,  the  cost  of  the  watch  was  either  $30  or  $70 ;  for  either  of 
these  answers  satisfies  the  conditions  of  the  problem. 

2.  A  farmer  bought  some  sheep  for  $72.  If  he  had 
bought  6  more  for  the  -same  money,  they  would  have  cost 
him  $  1  apiece  less.     How  many  did  he  buy  ? 


QUADRATIC   EQUATIONS.  239 

Let  X  =  the  number  bought. 

72 
Then,  —  =  the  number  of  dollars  paid  for  one, 

X 

72 

and =  the  number  of  dollars  paid  for  one  if  there 

X  4-  6 
^  had  been  6  more. 

72  72 

By  the  conditions,     —  =  +  1. 

X      x  +  6 

Solving,  X  =  18  or  -  24. 

Only  the  positive  value  of  x  is  admissible,  for  the  negative  value 
does  not  satisfy  the  conditions  of  the  problem. 
Therefore,  the  number  of  sheep  was  18. 

Note  1.  In  solving  problems  which  involve  quadratics,  there  will 
usually  be  two  vahies  of  the  unknown  quantity;  and  those  values  only 
should  be  retained  as  answers  which  satisfy  the  conditions  of  the 
problem. 

Note  2.  If,  in  the  enunciation  of  the  problem,  the  words  "6 
more"  had  been  changed  to  "6  fewer,''''  and  "$1  apiece  less"  to 
"SI  apiece  wore,"  we  should  have  found  the  answer  24. 

In  many  cases  where  the  solution  of  a  problem  gives  a  negative 
result,  the  wording  may  be  changed  so  as  to  form  an  analogous  prob- 
lem to  which  the  absolute  value  of  the  negative  result  is  an  answer. 

3.  I  bought  a  lot  of  flour  for  $126;  and  the  number 
of  dollars  per  barrel  was  ^  the  number  of  barrels.  How- 
many  barrels  were  purchased,  and  at  what  price? 

4.  Divide  the  number  18  into  two  parts^  the  sum  of 
whose  squares  shall  be  170. 

5.  Find  two  numbers  whose  difference  is  7,  and  whose 
sum  multiplied  by  the  greater  is  400. 

6.  Find  three  consecutive  numbers  whose  sum  is  equal 
to  the  product  of  the  first  two. 

7.  Divide  the  number  20  into  two  parts  such  that  one 
is  the  square  of  the  other. 

8.  Find  two  numbers  whose  sum  is  7,  and  the  sum  of 
whose  cubes  is  133. 


240  ALGEBRA. 

9.  Find  four  consecutive  numbers  such  that  if  the  first 
two  be  taken  as  the  digits  of  a  number,  that  number  is 
equal  to  the  product  of  the  other  two. 

10.  A  merchant  bought  a  quantity  of  flour  for  $  108.  If 
he  had  bought  9  barrels  more  for  the  same  money,  he  would 
have  paid  $  2  less  per  barrel.  How  many  barrels  did  he 
buy,  and  at  what  price  ? 

11.  A  farmer  bought  a  number  of  sheep  for  ^378. 
Having  lost  6,  he  sold  the  remainder  for  $  10  a  head  more 
than  they  cost  him,  and  gained  $  42.  How  many  did  he 
buy? 

^ 12.   A  merchant  sold  a  quantity  of  wheat  for  $  56,  and 

gained  as  many  per  cent  as  the  wheat  cost  dollars.     What 
was  the  cost  of  the  wheat  ? 

13.  If  the  product  of  three  consecutive  numbers  be 
divided  by  each  of  them  in  turn,  the  sum  of  the  three 
quotients  is  74.     What  are  the  numbers? 

14.  A  crew  can  row  8  miles  down  stream  and  back  again 
in  4f  hours ;  if  the  rate  of  the  stream  is  4  miles  an  hour, 
find  the  rate  of  the  crew  in  still  water. 

15.  A  certain  farm  is  a  rectangle,  whose  length  is  three 
times  its  width.  If  its  length  should  be  increased  by  20 
rods,  and  its  width  by  8  rods,  its  area  would  be  trebled.  Of 
how  many  square  rods  does  the  farm  consist  ? 

^  16.  A  man  travels  9  miles  by  train.  He  returns  by  a 
train  which  runs  9  miles  an  hour  faster  than  the  first,  and 
accomplishes  the  entire  journey  in  35  minutes.  Required 
the  rates  of  the  trains. 

17.  The  area  of  a  rectangular  field  is  216  square  rods, 
and  its  perimeter  is  60  rods.     Find  its  length  and  width. 

18.  At  what  price  per  dozen  are  eggs  selling  when,  if  the 
price  were  raised  5  cents  per  dozen,  one  would  receive  twelve 
less  for  a  dollar  ? 


QUADRATIC   EQUATIONS.  241 

19.  A  merchant  sold  goods  for  $  18.75,  and  lost  as  many 
per  cent  as  the  goods  cost  dollars.     What  was  the  cost  ? 

20.  A  man  travelled  by  coach  6  miles,  and  returned  on 
foot  at  a  rate  5  miles  an  hour  less  than  that  of  the  coach. 
He  was  50  minutes  longer  in  returning  than  in  going. 
What  was  the  rate  of  the  coach  ? 

21.  A  square  picture  is  surrounded  by  a  frame.  The  side 
of  the  picture  exceeds  by  an  inch  the  width  of  the  frame ; 
and  the  number  of  square  inches  in  the  frame  exceeds  by  124 
the  number  of  inches  in  the  perimeter  of  the  picture.  Find 
the  area  of  the  picture,  and  the  width  of  the  frame. 

22.  The  circumference  of  the  fore-wheel  of  a  carriage  is 
less  by  4  feet  than  that  of  the  hind-wheel.  In  travelling 
1200  feet,  the  fore- wheel  makes  25  revolutions  more  than 
the  hind-wheel.     Find  the  circumference  of  each  wheel. 

.  23.  A  tank  can  be  filled  by  two  pipes  running  together 
in  3|  hours.  The  larger  pipe  by  itself  will  fill  it  sooner 
than  the  smaller  by  4  hours.  What  time  will  each  pipe 
separately  take  to  fill  it  ? 

24.  The  telegraph  poles  along  a  certain  railway  are  at 
equal  intervals.  If  there  were  two  more  in  each  mile,  the 
interval  between  the  poles  would  be  decreased  by  20  feet. 
Find  the  number  of  poles  in  a  mile. 

25.  A  and  B  gained  in  trade  $  2100.  A's  money  was  in 
the  firm  15  months,  and  he  received  in  principal  and  gain 
{^3900.  B's  money,  which  was  $5000,  was  in  the  firm  12 
months.     How  much  money  did  A  put  into  the  firm  ? 

26.  If  $  2000  amounts  to  f  2205,  when  put  at  compound 
interest  for  two  years,  the  interest  being  compounded  annu- 
ally, what  is  the  rate  per  cent  per  annum  ? 

27.  A  man  travelled  105  miles.  If  he  had  gone  4  miles 
more  an  hour,  he  would  have  performed  the  journey  in  9 J 
hours  less  time.     How  many  miles  an  hour  did  he  go  ? 


242  ALGEBRA. 

28.  The  sum  of  $120  was  divided  be  ween  a  certain 
number  of  persons.  If  each  person  had  received  f  7  less, 
he  would  have  received  as  many  dollars  as  there  were 
persons.     Eequired  the  number  of  persons. 

29.  My  income  is  $  5000.  After  deducting  a  percentage 
for  income  tax,  and  then  a  percentage,  less  by  one  than  that 
of  the  income  tax,  from  the  remainder,  the  income  is  reduced 
to  $  4656.     Find  the  rate  per  cent  of  the  income  tax. 

30.  A  man  has  two  square  lots  of  unequal  size,  together 
containing  13,325  square  feet.  If  the  lots  were  contiguous, 
it  would  require  510  feet  of  fence  to  embrace  them  in  a 
single  enclosure  of  six  sides.     Find  the  area  of  each  lot. 

31.  A  merchant  has  a  cask  full  of  wine,  containing  36 
gallons.  He  draws  a  certain  number  of  gallons,  and  then 
fills  the  cask  up  with  water.  He  then  draws  out  the  same 
number  of  gallons  as  before,  and  finds  that  there  are  25 
gallons  of  pure  wine  remaining  in  the  cask.  How  many 
gallons  did  he  draw  each  time  ? 

32.  A  set  out  from  C  towards  D  at  the  rate  of  5  miles  an 
hour.  After  he  had  gone  32  miles,  B  set  out  from  D  towards 
C,  and  went  every  hour  Jy  of  the  entire  distance ;  and  after 
he  had  travelled  as  many  hours  as  he  went  miles  in  an 
hour,  he  met  A.     Eequired  the  distance  from  C  to  D. 

33.  A  courier  travels  from  P  to  Q  in  12  hours.  Another 
courier  starts  at  the  same  time  from  a  place  24  miles  the 
other  side  of  P,  and  arrives  at  Q  at  the  same  time  as  the 
first  courier.  The  second  courier  finds  that  he  takes  half 
an  hour  less  than  the  first  to  accomplish  12  miles.  Find 
the  distance  from  P  to  Q. 

34.  A  man  bought  a  number  of  $50  shares,  when  they 
were  at  a  certain  rate  per  cent  premium,  for  $4800;  and 
afterwards,  when  they  were  at  the  same  rate  per  cent  dis- 
count, sold  them  all  but  30  for  $  2000.  How  many  shares 
did  he  buy,  and  how  much  did  he  give  apiece  ? 


QUADRATIC  EQUATIONS.  243 


XXIII.  EQUATIONS  SOLVED  LIKE  QUAD- 
RATIOS. 

EQUATIONS  IN  THE  QUADRATIC  FORM. 

269.  An  equation  is  said  to  be  in  the  quadratic  form  when 
it  is  expressed  in  three  terms,  two  of  which  contain  the 
unknown  quantity,  and  the  exponent  of  the  unknown  quantity 
in  one  of  these  terms  is  twice  its  exponent  in  the  other;  as, 

x^-\-x^  =  72;  etc. 

270.  Equations  in  the  quadratic  form  may  be  readily 
solved  by  the  rules  for  quadratics. 

1.  Solve  the  equation  a^  —  6ar^  =  16. 

Completing  the  square  by  the  rule  of  §  258, 

jc«-6ic8_|_32  =  16  +  9  =  25. 
Extracting  the  square  root,       a;*  —  3  =  ±  5. 
Whence,  x^  =  3  ±  5  =  8  or  -  2. 

Extracting  the  cube  root,  x  =  2  or  —  V2,  Ans. 

Note  1.  There  are  also  four  imaginary  roots,  which  may  be  ob- 
tained by  the  method  of  §  267. 

2.  Solve  the  equation  2x  -{-  3V^  =  27. 

Since  Vx  is  the  same  as  a;*,  this  is  in  the  quadratic  form. 
Multiplying  by  8,  and  adding  S^  to  both  members  (§  261), 

16x  +  24  Vx  +  32  =  216  +  9  =  225. 

Extracting  the  square  root,  4a/x  +  3  =  ±  15. 

4Vx  =  -3±15  =  12or  -18. 

Whence,  Vx  =  3  or 

2 

81 
Squaring,  x  =  9  or  — ,  Ans. 

4 


244  ALGEBRA. 

3.    Solve  the  equation  16  x~^^  -  22  x~^  =  3. 
Multiplying  by  16,  and  adding  IP  to  both  members, 

162x"^  -  16  X  22  cc"^  +  112  =,  48  +  121  =  169. 

_.i 
Extracting  the  square  root,  16  a;  ^  —  11  =  ±  13. 


_3  13 

Whence,  x  ^  =  —  or  - 


16ic"?  =  ll±13  =  -2or24. 

lor§. 

8       2 


Extracting  the  cube  root,  x  ^  =  —  or  I -\  . 

—1  1  /^\3^ 

Raising  to  the  fourth  power,  x     =  —  or  (  ^  j  • 

Inverting  both  members,  x  =  16  or  (- ],  Ans. 

Note  2.  In  solving  equations  of  the  form  x^  =  a,  first  extract  the 
root  corresponding  to  the  numerator  of  the  fractional  exponent,  and 
then  raise  to  the  power  corresponding  to  the  denominator.  Particular' 
attention  should  be  paid  to  the  algebraic  signs ;  see  §§186  and  193. 

EXAMPLES. 
Solve  the  following  equations : 

4.  35^-21x2  = -108.  8    12a?-2  4-a^-i  =  35. 

5.  8a;4-14V^  =  15.  9-   a5*  +  a;*  =  702. 

6.  0^-30^^  =  88.  10.   16o^-«-30o.'-«=81. 


7.   32o.'«  +  -.  =  -33. 


^  '     Vi 


X 


12.  (2o^- 3)2  =  -260^3  +  153. 

13.  (5  x-^  -  2)2  -  16(o;-2  + 1)^  =  -  76. 

14.  9x''*-22o;"^  =  -8.  17.   o;»  -  97 o.-^  + 1296  =  0. 

15.  3.?  +  2.?  =  16.  ^g    3.t-^  =  94. 

16.  x-^-Ux-^  =  -225.  as* 


QUADRATIC   EQUATIONS.  245 

19.   4a;"?+27a;-«  =40.  22.   2  x~' -^  59  x'^  =  160. 

23    ^  "^  ^  —  ^  +  ^ 

21.   27 V^^  +  lOv  ^  =  128.  *     V^  ~   V^  ' 


24.    V5+V^  +  -vVWi  = 


V5+v« 


25.    ^3  +Vx  +^4:  -Vx  =^7  +  2-y/x. 

271.   Au  equation  may  sometimes  be  solved  with  refer- 
ence to  an  expression,  by  regarding  it  as  a  single  quantity. 

1.  Solve  the  equation  (x  —  5y  —  S(x  —  5)^  =  40. 
Multiplying  by  4,  and  adding  3^  to  both  members, 

4(0;  -  5)8  -  12(x  -  5)^  +  32  =  160  +  9  =  169. 

Extracting  the  square  root,  2 (x  —  6)^—  3=  ±  13. 

2(x  -  5)^  =  3  ±  13  =  16  or  -  10. 

Whence,  (x  -  5)^  =  8  or  -  5. 

Extracting  the  cube  root,  (x  -  6)2  =  2  or  —  v^S. 

Squaring,     •  x  -  5  =  4  or  v^25. 

Whence,  x  =  9  or  5  +  \/25,  Ans. 

Certain  equations  of  the  fourth  degree  may  be  solved  by 
the  rules  for  quadratics. 

2.  Solve  the  equation  a;*  +  12  a:^  +  34  ar^  -  12  a;  -  35  =  0. 

The  equation  may  be  written 

(x*  +  12  x8  +  36x2)  -  2  x2  -  12  X  =  35. 
That  is,  (x2  +  6  x)2  -  2  (x2  +  6  x)  =  35. 

Completing  the  square, 

(x2  +  6x)2  -  2(x2  +  6x)+ 1  =  36. 
Extracting  the  square  root, 

(x2  +  6x)-  1  =±6. 

x2  +  6x  =  l  ±6  =  7  or  -5. 


246  ALGEBRA. 

Completing  the  square,     x-  +  6  x  +  9  =  16  or  4. 
Extracting  the  square  root,         a;  +  3=±4or  ±2. 
Whence,  x  =  -3±4or  -3±2 

=  1,  —  7,  —  1,  or  —  5,  A?is. 

Note  1.  In  solving  equations  like  the  above,  we  first  form  a  per- 
fect square  witli  the  x*  and  x^  terms,  and  a  portion  of  the  x^  term. 
By  §  258,  the  third  term  of  the  square  is  the  square  of  the  quotient 
obtained  by  dividing  the  x^  term  by  twice  the  square  root  of  the  x* 
term. 


3.    Solve  the  equation  x^  —  6x-\-  ^ym?  —  6 a;  -f  20  =  46. 
Adding  20  to  both  members, 


(x2  -  6 X  +  20)  +  5 Vx2-6x  +  20  =  66. 
Multiplying  by  4,  and  adding  5^  to  both  members. 


4(x2  -  6x  +  20)+  20Vx2^  6x  +  20  +  52  =  264  +  25  =  289. 
Extracting  the  square  root, 


2Vx2-6x  +  20  +  5=:±17. 


2Vx2-6x  +  20  =  -  6  ±  17  =  12  or  -  22. 


Whence,  Vx^  -  6  x  +  20  =  6  or  -  11. 

Squaring,  x2  -  6  x  +  20  =  36  or  121. 

Completing  the  square,  x2  —  6  x  +  9  =  25  or  110. 

Extracting  the  square  root,  x  —  3  =  ±5or  ±  vTIO. 

Whence,  x  =  8,  -  2,  or  3  ±  VUO, 

-4  ns. 

Note  2.  In  solving  equations  like  the  above,  add  such  a  quantity 
to  both  members  that  the  expression  without  the  radical  in  the  first 
member  may  be  the  same  as  that  within,  or  some  multiple  of  it. 

EXAMPLES. 
Solve  the  following  equations : 

4.  (a^-2a;)2-18(»2-2a;)  =  -45. 

5.  x^  +  8a^-10x2-104a;4-105  =  0. 

6.  a;^-10x'«  +  23x2  4-10«-24  =  0. 


If, 

QUADRATIC   EQUATIONS.      '  247 


7.  x'-\-7  +  Vx''-\-7  =  20. 

8.  V3x-2-5-^3x-2  =  -6. 


9.'  x^-2  a;  -t-  OVx"  -  2  a;  +  5  =  11. 

10.  a^  +  2a;  +  3  =  Va^  +  2a;  +  9. 

11.  </3^~^^^2^-'s/3x-2x'  =  2. 

12.  (ar^  +  17)^  _  35  (ar^  +  17)*  =  -  216. 

13.  (2  a; +  5)-^ +  31  (2  a; +  5)-'^  =  32. 

14.  x'-\-Ua^-\-71x'-^WAx -{-120  =  0. 


15.  2«2-3a;H-6V2a^-3a;  +  2  =  14. 

16.  4a;^-12a:3  4-7a^  +  3a;-2  =  0. 

17.  (3ar^  +  a5-l)3-26(3ar'  +  a;-l)^  =  27. 


18.  4a^-9a;  +  23  =  7V4F^-^9^Tli. 

19.  (x'-\-by  =  2aa^-\-2ab'x-a^x'. 

20.  (a;-a)^-56(a;-a)^  +  662  =  0. 


y  21.  2(ar^-2a;)4-3Var^-2a;4-6  =  15. 

22.  3aj2-9a;  =  4Var»-3a;4-5-ll. 

23.  8(5a;-3)-^-6(5a;-3)-^  =  -l. 

24.  2(2a.'3  +  10)-^+3(2a;3  +  10)-^=2. 

26.  x'-^4:ax^-34:a^x'-76a'x-\-105a*  =  0. 


248  ALGEBRA. 

XXIV.   SIMULTANEOUS  EQUATIONS. 

INVOLVING  QUADRATICS. 

272.  An  equation  containing  two  unknown  quantities  is 
said  to  be  symmetrical  with  respect  to  them  when  they  can 
be  interchanged  without  destroying  the  equality. 

Thus,  the  equation  a^  —  xy  -\- y^  =  3  is  symmetrical  with 
respect  to  x  and  y ;  for  on  interchanging  x  and  y,  it  becomes 
y^  —  yx  -{-  ay^  =  3,  which  is  equivalent  to  the  first  equation. 

But  the  equation  x  —  y  =  1  is  not  symmetrical  with 
respect  to  x  and  y ;  for  on  interchanging  x  and  y,  it  becomes 
y—x  =1,  OT  X  —  y  =  —  1,  which  is  a  different  equation. 

273.  An  equation  containing  two  unknown  quantities  is 
said  to  be  homogeneous  when  the  terms  containing  the 
unknown  quantities  are  of  the  same  degree  with  respect 
to  them  (§  157). 

Thus,  the  equation  op^  —  3xy  —  2y^=l  is  homogeneous, 
for  the  terms  containing  a;  and  ?/  are  of  the  second  degree 
with  respect  to  x  and  y. 

But  the  equation  a^  —  2  ?/  =  3  is  not  homogeneous  ;  for  x^ 
is  of  the  second  degree,  and  2y  of  the  first  degree. 

274.  On  the  use  of  the  double  signs  ±  and  =F . 

If  two  or  more  equations  involve  double  signs,  it  will  be 
understood  that  the  equations  can  be  read  in  two  ways ; 
first,  reading  all  the  2ipper  signs  together ;  second,  reading 
all  the  lower  signs  together. 

Thus,  the  equations  x  =  ±2,  y  =  ±  3,  can  be  read  either 
a;  =  +  2,  2/  =  +  3,  or  a^  =  -  2,  2/  =  -  3. 

Also,  the  equations  x  =  ±2,  ?/  =  q:  3,  can  be  read  either 
a;  =  -h2,  y  =  -3,  or  x  =  -2,  y  =  -\-3. 


SIMULTANEOUS   EQUATIONS.  249 

275.  Two  equations  of  the  second  degree  (§  158)  with 
two  unknown  quantities  will  generally  produce,  by  elimina- 
tion, an  equation  of  the  fourth  degree  with  one  unknown 
quantity ;  the  rules  already  given  are,  therefore,  not  suffi- 
cient to  solve  all  cases  of  simultaneous  equations  of  the 
second  degree  with  two  unknown  quantities. 

Consider,  for  example,  the  equations 

rar^-f2/=5.  (1) 

\x+f  =  S.  (2) 

From  (1),  y  =  5-s^. 

Substituting  in  (2),  a;  -f  25  -  10  x-  -f-  x*  =  3 ; 
which  is  an  equation  of  the  fourth  degree. 

In  several  cases,  however,  the  solution  may  be  effected 
by  means  of  the  rules  for  quadratics. 


276.   Case  I.      When  each  equation  is  in  the  form 

a^-\-by^  =  c. 

1.   Solve  the  equations 

f3r^-f    4r'  =  76. 

(1) 
(2) 

Multiplying  (1)  by  3,                  9  x^  +I2y2  =  228. 

Multiplying  (2)  by  4, 
Subtracting, 

12  2/2-44/2=    16. 

53x2  =  212. 

Then,                                                         x^  =  4,  and  x  =  ±2. 

(3) 

Substituting  from  (3)  in  (1), 

12  -h  4  y2  =  76. 

4  2/2  =  64. 
Then,  y-2  =16,  and  y  =  ±4. 

Ans.  X  =  2,  y  =  ±4;  or,  x  =  — 2,  y  =  ±i. 

Note.  In  this  case  there  are  four  possible  sets  of  values  of  x  and 
y  which  satisfy  the  given  equations  : 

1.  x  =  2,  2/  =  4.  3.    x  =  -2,  j/  =  4. 

2.  x  =  2,  y  =  -4.  4.    a;  =  -2,  !/=-4. 

It  would  be  incorrect  to  leave  the  result  in  the  form  x  =  ±  2, 
y  =  ±i;  for,  by  §  274,  this  represents  only  the  first  and  fourth  of 
the  above  sets  of  values. 


250  ALGEBRA. 

EXAMPLES. 

Solve  the  following  equations : 

2x'^Sy'  =  93.  '     [12  0)2 +  13/ =  248. 


277.  Case  II.  When  one  equation  is  of  the  second  degree, 
and  the  other  of  the  first. 

Equations  of  this  kind  may  always  be  solved  by  finding 
the  value  of  one  of  the  unknown  quantities  in  terms  of  the 
other  from  the  simple  equation,  and  substituting  this  value 
in  the  other  equation. 

1.    Solve  the  equations    \  " '    ~^^  ~  ^  \J 

1    x  +  2y  =  l.  (2) 

From  (2),  2y  =  l  -x,ovy=  l^:^.  (3) 

Substituting  in  (1),  2^2  -  a^(^^^)  =  qH-^^Y 

Clearing  of  fractions,  4  x^  —  7  x  +  x^  =  42  —  6  x. 

Or,  6x2-x  =  42. 

14 
Solving  this  equation,  aj  =  3  or  —  — - 

5 

7  +  11 
7  —  3  5  49 

Substituting  in  (3),  y  =  — -—  or  — ^ — =  2  or  — • 

14  49 

X  =  3,  y  =  2  ;  or,  x  =-—,«/  =  —,  Ans. 
5  10 

Note.  Certain  examples  where  one  equation  is  of  the  third  degree, 
and  the  other  of  the  first,  may  be  solved  by  the  method  of  Case  II. 


EXAMPLES. 
Solve  the  following  equations  : 

\2x+y  =  l. 


x  +  y  =  -S. 
xy  —  —  54. 


^ 


MUI/r  \\F,<  M  -    EQUATIi 


\MPLES. 

luatioiis : 


12 

t 

13.    I 


14     . 


I  ^-'  4- 1/«  =  260. 
i  .r  — //  =  -  14. 

a^  =  -  80. 
.r-y  =  24. 

.r"4-/  =  .504. 


1  dj*  —  xy 


15     i 


16 


jc^  '—  xy  -i-  u 


r  —  7/  =    - 


degree,  and  the  other  of  the  second  or  first.    . 

Equations  of  tliis  kind  may  always  be  solved  by  combin- 
ing them  in  such  a  way  as  to  obtain  the  values  oi  x  -\-y 
and  x  —  y. 


1.   Solve  the  equations 


'  ic  -f  2/  =  2. 
xy  =  — 15. 

a;2  +  2xy  +  2/^  =  4. 

Axy  =  - 


(1) 
(2) 


Squaring  (J), 

Multiplying  (2)  by  4, 

Subtracting,  x^  —  2xy  +  y^  =     64. 

Extracting  the  square  root,  x  —  y  =  ±S.                       (3) 

Adding  (1)  and  (3),  2  x  =  2  ±  8  =  10  or  -  6. 

Whence,  2c  =  6  or  —  3. 

Subtracting  (3)  from  (1) ,  2  y  =  2  ^F  8  =  -  6  or  10. 

Whence,  y  =  —  3  or  5. 

a;  =  5,  1/  =  —  3  ;  or,  x  =  —  3,  y  =  5,  Ans. 


2oO 


m 


EXAr, 


4. 
,^-  =  93. 

277.   L'.^.-.^  ..  tinu  is  nf  the  secoml  (hyif^'. 

aii/l  the  other  of  the/ 

this  kind  may  always  be  solved  by  limlifig 
.<  n.«  .,.,1  ,,....      f|nantities  in  terms  <^^f  '»>" 
oii  nd  substituting  tliift,. 


a;  =  4  or  —  2.  '    "' 

Subtracting  (3)  from  (6),  2  ?/  =  ±  6  -  2  =  4  or  -  8. 

Whence,  ?/  =  2  or  —  4. 

X  =  4,  y  =  2;  or,  x  =  —  2,  ?/  =  —  4.  ^ws. 

Note  3.  The  above  equations  are  not  symmetrical  according  to 
the  definition  of  §  272  ;  but  the  method  of  Case  III.  may  often  be  used 
in  cases  where  the  given  equations  are  symmetrical  except  with  respect 
to  the  signs  of  the  terms. 

x'-^-f-^  50.  (1) 

xy  =  -7.  (2) 

Multiplying  (2)  by  2,  2xy  =  -  14.  (3) 

Adding  (1)  and  (3),  x'^-^2xy  +  y^  =  36. 

Whence,  x-{-  y  =  ±Q-  (4) 

Subtracting  (3)  from  (1),  x^  -2xy  +  y^  =^64. 
Whence,  x  —  y  =  ±S.  (6) 

Adding  (4)  and  (5),  2x  =  G±  8,  or  -6±  8. 

Whence,  x  =  7,  —  1,  1,  or  —  7. 

Subtracting  (5)  from  (4),  2  y  =  6  T  8,  or  -  6  T  8. 

Whence,  y  =  —  1,  7,  —  7,  or  1. 

«  =  ±  7,  y  =  T  1 ;  or,  X  =  ±  1,  y  =  T  7,  Ans, 


(  X^  -\-y-  =  , 

3.    Solve  the  equations  \ 

[xy  =  -7. 


SIMULTANEOUS   EQUATIONS. 


253 


EXAMPLES. 

e  the  following  equations  : 
.'2/ =  48. 
ic  -|-  2/  =  14. 

ar^-f  2/2=101. 
a;  +  2/  =  -  9. 

f.T3_2/-^  =  37.  " 

[:^-^xy  +  if  =  Sl. 

|.r2/  =  45. 
yx  —y  =  —  4. 

f  a^  =  12. 
lar^  +  2/^  =  40. 
a;3_2/3^133 

x-y  =  l. 

a5»  +  2/«  =  ^17. 

«"  +  ajy  +  ?/"  =  39. 

X  -h  y  =  -  2. 


12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 


r  .r2  +  2/'  =  260. 
(^  .r  —  2/  =  —  14. 
f  iB2/  =  -  80. 

|..'-y=:24. 

{^Jrf  =  504. 
j  a^  _  a;2/  +  2/-  -  84. 

I  ar^  -  an/  +  2/'  =  63. 

|a;-2/  =  -3. 

.^2  +  2/'  =  305. 
a; -2/ =  21. 

a^H- 2/2^218. 
a^  =•-  91. 

a!»4-3^  =  -^35. 
a^  —  a^ -I- 2/2  =  67. 

«2/  =  - 150. 
a;  -  y  =  -  31. 


279.   Case  IV.     When  each  equation  is  of  the  second  degree, 
■I  ■  omogeneous  (§  273). 

"-.e  1.  Certain  equations  which  are  of  the  second  degree  and 
eiieous  may  be  solved  by  the  method  of  Case  I.  or  Case  III. 
-X.  1,  §  276,  and  Kx.  .3,  §  278.) 

method  of  Case  IV.  should  be  used  only  when  the  example 
!-  be  solved  by  the  methods  of  Cases  I.  or  III. 


I     Solve  the  equations 


aj2  —  2  a;?/  =  5. 
ar^  4-      2/'  =  29. 
lilting  y  =  vx  in  the  given  equations,  we  have 

«2_2vx2  =  5;  or,  x^ 


x2  +  v^x^  =  29  ;  or,  x^ 


1  -2v' 
29 


(1) 


1  +t?2 

Divide  the  first  equation  by  the  second. 


254 

Equating  the  values  of  x^, 

Or, 
Or, 

Solving  this  equation, 


ALGEBRA. 

5  29 


1  -  2 1;      1  +  ^2 
5  +  5v2  =  29-58v. 
5  u2  +  58 1?  =z  24. 

12. 


V  =-  or 
5 


Substituting  these  values  in  (1),     x^ 


5  5 


l+24     ^^"^25- 


Whence, 


X  =  ±  5  or  i: 


V5 


Substituting  the  values  of  v  and  x  in  the  equation  y  =  vx, 


If  v  =  -  andiK  =  ±5,  ?/=?(±5)  =  ±2. 
6  o 


If  v  =  -12  and  ^  =  ±^'  2/  =  -12^±^^  =  T 


12  V5 


iN'ote  2.  In  finding  ?/  from  the  equation  y  =  vx,  care  must  be 
taken  to  multiply  each  pair  of  values  of  x  by  the  corresponding  value 
otv. 


EXAMPLES 
Solve  the  following  equations  : 

(2x^-xy  =  2S. 
1  x'-{-2y'  =  lS. 

(  x^  -{-  xy  =  —  6. 


6. 


7. 


[xy-y^  =  -  35. 

fc^-^xy-\-y'  =  6S. 

1  a^  -  2/2  =  -  27. 

1^2 +  32/' =  28. 

I  a^  4- a;?/ +  2/=  16. 

-  a;2  -  2  a;2/  =  84. 
.  2  a;^/  —  2/'  =  —  64. 


8. 

9. 
10. 
11. 


3a52-}-a;2/-32/'  =  33. 
2a^-2/2  =  23. 

x^-{-  6  xy  —  2/2 


7. 


x^-^3xy-2y^  =  -4:. 

(x^-xy-12y'  =  S. 
[af^-\-xy-10y'  =  20. 

5x^-4.xy=33. 
27x'-32xy-4:y'=55. 

|3aj2  +  a?2/  +  2/2  =  47. 
[4.x^-3xy-y^  =  -39. 


SIMULTANEOUS   EQUATIONS.  255 

MISCELLANEOUS  AND   REVIEW  EXAMPLES. 

280.  No  general  rules  can  be  given  for  the  solution  of 
examples  wliich  do  not  come  under  the  cases  just  considered. 
Various  artifices  are  employed,  familiarity  with  which  can 
only  be  gained  by  experience.  ' 

x^-f^  19.  (1) 

a^-xy^  =  6.  (2) 

Multiplying  (2)  by  3,  Sx'^y  -  Sxy^  =  18.  (3) 

Subtracting  (.3)  from  {I),  x^  -  Sx^y -\- ^xtf-  -  y^  =  I. 
Extracting  the  cube  root,  x  —  y  =  1.  (4) 

Dividing  (2)  by  (4),  xy  =  6.  (5) 

Solving  equations  (4)  and  (5)  by  the  method  of  Case  III.,  we  find 
X  =  S,  y  =  2  ;  or,  x  =  -  2,  y  =  -  3,  Ans. 


1.   Solve  the  equations    | 


(  Qir^  -\- y^  =  9  xy. 
2.   Solve  the  equations  \ 

[x-\-y  =  6. 


Putting  X  =  ?(  +  V  and  y  =  u  —  v,  vre  have 

(M  +  vy  +  (u  -  vy  =  9(n+v)(ti-v),  (1) 
and  (M+t>)  +  (M-v)=6.  (2) 

Keducing  (1),  2 «»  +  6  uv^  =  9(m2  -  v^).  (3) 

Reducing  (2),  2  m  =  6,  or  u  =  3. 

Substituting  the  value  of  u  in  (3),  54  +  18«2  =  9(9  -  v^). 

Whence,  u^  =  1,  or  «  =  ±  1. 

Therefore,  x  =  M  +  v  =  3±l=4or2, 

and  ?/  =  «-v  =  3=Fl  =  2or4. 

X  =  4,  y  =2;   or,  x  =  2,  r/  =  4,  Ans. 

Note.  The  artifice  of  substituting  u  +  v  and  u  -  v  for  x  and  y  is 
applicable  in  any  case  where  the  given  equations  are  symmetrical  with 
respect  to  x  and  y  (§  272).     See  also  Ex.  4,  p.  256. 

(x'^f  +  2x  +  2y  =  2S.  (1) 

3.    Solve  the  equations   ]  ^  ,^. 

[  xy  =  6.  (^) 

Multiplying  (2)  by  2,  2xy  =  12.  (3) 

Adding  (1)  and  (3),  x"^ +  2xy  +  y^- +  2x-{-2y=  35. 


256  ALGEBRA. 

Or,  (a;  +  ,y)2  +  2(x  +  y)=35. 

Completing  the  square,  (x+y)24-2(x +«/)  +  !=  36. 

Whence,  (x  +  y)  +  1  =  ±  6, 

or  X  +  ?/  =  5  or  —  7.              (4) 

Squaring  (4),  x^  +  2  xj/ +  2/^  =  25  or  49. 

Multiplying  (2)  by  4,  ixy          =  24, 

Subtracting,  x^  —  2xy  +  y^=  1  or  25. 

Whence,  x  —  y  =  i  1  or  ±  5.          (5) 

Adding  (4)  and  (5),  2x  =  5  ±  1,  or  -  7  ±  5. 

Whence,  x  =  3,  2,   -  1,  or  -  6. 

Subtracting  (5)  from  (4),  2?/  =  5  T  1,  or  -7^5. 

Whence,  ?/  =  2,  3,   -  6,  or  -  1. 

x=3,  y  =  2;  x=2,  y=S;  x=— 1,  y=—Q;  or,  x=-6,  ?/=  -  1,  ^ws. 


4.    Solve  the  equations    ]  " 

[    x-\-y  =  -l. 


Putting  X  =  u  -{■  V  and  y  =  u  —  -y,  we  have 

(u-^vy-{-(u-vy  =  97,  (1) 

and  (u  +  v)  +  {u-v)  =  -l.  (2) 

Reducing  (1),       2  Jt*  +  12  ?<'V-^  +  2  v*  =  97.  (3) 

Reducing  (2),  2  w  =  -  1,  or  m  =- -. 

Substituting  in  (3) ,       ^  +  3  ?j2  +  2  v*  =  97. 
8 

Solving  this  equation,  v^  =  —  or  — -• 

4  4 

Whence,                                              v  =  ±  -  or  ±      ~      • 
Then,  

and 

1^5            1  ^  V^niT        o    „    „,  -1  tV-31 
j,  =  «_«  =  _-T-  or  --T^-  =  -3,  2,  or  ^ _ 

x=2,  2,=  -3;  a;=-3,  y=2  ;  or,  ^^-1±V-3T     y^-lTV^^, 


SIMUI^ANEOUS   EQUATIONS. 


257 


EXAMPLES. 
Solve  the  following  equations  : 


12 

h'^  =  -25- 

r  car  +  jf  ^x-y  =  26. 
\xy:=12.       • 

(2x'-3xy  =  -4:. 
\  4  xy  —  5y^  =  3. 

1 4  ic2  _  5  ^.y  ^  19 

I  ^jy  +  /  =  6. 
1-1  =  1. 

a;     2/      2 


18* 


a^  +  2  /  =  47  +  2  a;, 
a.-^  -  2  2/2  =  _  7. 

jj     p  +  a^  +  2/2  =  97. 
\x-y  =  19. 

(x^-\-f  =  756. 
a.-2  -  xy  +  2/'  =  63. 

xY  +  28  .i-2/  ~  480  =  0. 
2x  +  y=:ll. 


10 


12. 
13. 


14. 


15. 


i  +  i  =  A. 

x'     f     16 


1_1 

a;     2/ 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


[x-\-y=l. 


25. 


26. 


(x^-y^  =  3. 

(a^-\-4:y'-\-3x  =  22.  ■ 
\2xy-^3y-\-9  =  0. 

3x^-5xy-\-2f  =  -3 
4:  X  —  oy  =  10. 

i.  xy  =  a^  —  1. 

[  a;  -h  2/  =  2  a. 

'^+?=^^- 

a;     y 

x-\-y     x  —  y^lO 
x  —  y     X  -{-y      3 
I  ar^  +  2/^  =  45. 

.^-h2/'  =  2a3  4-6a62. 
a.'y(a;  +  2/)=2a='-2a6l 

2x"2-3a^=15a-10a2. 
3;r  +  22/=12a-13. 

x^  -f  a^2/'  +  2/'  =  91. 
x^  +  xy  +  y^  =  13. 

x'  +  f  =  13(a'  +  l). 
x-\-y  =  ^a  —  l. 

2ar^+3x2/-42/'=-20. 
5aj2-72/'=-8. 


*  Divide  the  first  equation  by  the  second. 


268 


ALGEBRA. 


28. 


27. 


x'^xy  +  y^  =  3a^-Sab-^3b\ 


y  =  ^• 


29.  \^^^-^y-y  =  ^^- 

[  —5xy-\-y^-\-3x  = 


81. 


30. 


31. 


32. 


33. 


34. 


35. 


36. 


-\/x 


2/ =  19. 

2         7 


y      X         2 
x-\-y  =  l. 


aj  -  2/  =  1. 

f  x^y  —  ic  =  —  14. 
I  .'cy  +  aj2  =  148. 

x^  —  xy  =  27  2/. 
ajy  —  2/^  =  3  ic. 


x-\-y  2x—y  ^15 
X  —  y  x-{-  2y  4 
x-3y  =  -2. 

y(x  —  a)=2ab. 
x(y  —  b)=2  ab. 


37. 
38. 
39. 
40. 

41. 
42. 
43. 
44. 


=  -T-'       45. 


46. 


(xP-y'  = 
[x-y=l 

X^  =  X  -{-  I 

f  =  Sy- 


31. 


I  Va;2  +  7  =  6-2/. 

I  V^^+227  =  22  -  ic2. 

5SxP-12Sxyi-64.y^=5. 
26x^-62xy  +  32y'=:5, 

xy-(x-y)  =  l. 
xY  +  («  -  yf  =  13. 

i»2/+  C^'  — 2/)  =—5. 
xy{x-y)=-SL 

y?  —  xy  -\-y^  =  12. 
0,-3  _|_  2/3  _j_  3  a;^/  =  48. 

:^^rxy^y^  =  7. 
a:  +  2/  =  5  H-  a;?/. 

2a^  + 22/2  =  5  a;?/. 

ar^-2a;2/  +  3a;2  =  -16. 

2x-3y^l. 

3  a; +  5)3  =  -14. 


PROBLEMS. 
Note.     In  the  following  problems,  as  in  those  of  §  268,  only  those 
answers  are  to  be  retained  which  satisfy  the  given  conditions. 

281.  1.  The  sum  of  the  squares  of  two  numbers  is  52, 
and  their  difference  is  one-fifth  of  their  sum.  Find  the 
numbers. 

2.  The  difference  of  the  squares  of  two  numbers  is  16, 
and  their  product  is  15.     Find  the  numbers. 


SIMULTANEOUS  EQUATIONS.  259 

3.  If  the  length,  of  a  rectangular  field  were  increased  by 
2  rods,  and  its  width  diminished  by  5  rods,  its  area  would 
be  80  square  rods ;  and  if  its  length  were  diminished  by  4 
rods,  and  its  width  increased  by  3  rods,  its  area  would  be 
168  square  rods.     Find  its  length  and  width. 

4.  The  difference  of  the  cubes  of  two  numbers  is  218,  and 
the  sum  of  their  squares  is  equal  to  109  minus  their  prod- 
uct.    Find  the  numbers. 

5.  If  the  product  of  two  numbers  be  multiplied  by  their 
sum,  the  result  is  70 ;  and  the  sum  of  the  cubes  of  the  num- 
bers is  133.     Find  the  numbers. 

6.  A  farmer  bought  4  cows  and  8  sheef)  for  $  600.  He 
bought  5  more  cows  for  $490  than  sheep  for  $80.  Find 
the  price  of  each. 

7.  Find  a  number  of  two  figures  such  that,  if  its  digits  be 
inverted,  the  difference  of  the  number  thus  formed  and  the 
original  number  is  9,  and  their  product  736. 

8.  The  sum  of  two  numbers  exceeds  the  product  of  their 
square  roots  by  7;  and  if  the  product  of  the  numbers  be 
added  to  the  sum  of  their  squares,  the  result  is  133.  Find 
the  numbers. 

9.  The  sum  of  the  terms  of  a  fraction  is  13.  If  the 
numerator  be  decreased  by  2,  and  the  denominator  increased 
by  2,  the  product  of  the  resulting  fraction  and  the  original 
fraction  is  y\.     Find  the  fraction. 

10.  A  rectangular  mirror  is  surrounded  by  a  frame  3^ 
inches  wide.  The  area  of  the  mirror  is  384  square  inches, 
and  of  the  frame  329  square  inches.  Find  the  length  and 
width  of  the  mirror. 

11.  A  crew  row  up  stream  18  miles  in  4  hours  more  time 
than  it  takes  them  to  return.  If  they  row  at  two-thirds  their 
usual  rate,  their  rate  up  stream  would  be  one  mile  an  hour. 
Find  their  rate  in  still  water,  and  the  rate  of  the  stream. 


260  ALGEBRA. 

12.  A  rectangular  field  contains  2\  acres.  If  its  length 
were  decreased  by  10  rods,  and  its  width  by  2  rods,  its  area 
would  be  less  by  an  acre.     Find  its  length  and  width. 

13.  A  distributes  $  180  equally  between  a  certain  number 
of  persons.  B  distributes  the  same  sum  between  a  number 
of  people  less  by  40,  and  gives  to  each  $  6  more  than  A  does. 
How  many  persons  are  there,  and  how  much  does  A  give 
to  each  ? 

14.  A,  B,  and  C  together  can  do  a  piece  of  work  in  one 
hour.  B  does  twice  as  much  work  as  A  in  a  given  time; 
and  B  alone  requires  one  hour  more  than  C  alone  to  per- 
form the  work.  .In  what  time  could  each  alone  do  the  work  ? 

15.  If  the  length  of  a  rectangular  field  were  increased  by 
one-eighth  of  itself,  and  its  width  decreased  by  one-sixth  of 
itself,  its  area  would  be  decreased  by  60  square  rods,  and  its 
perimeter  by  2  rods.     Find  its  length  and  width. 

16.  If  the  product  of  two  numbers  be  added  to  their 
difference,  the  result  is  26;  and  the  sum  of  their  squares 
exceeds  their  difference  by  50.     Find  the  numbers. 

(Represent  the  numbers  hy  x  -h  y  and  x  —  y.) 

17.  A  sets  out  to  walk  to  a  town  21  miles  off,  and  one 
hour  afterwards  B  starts  to  follow  him.  When  B  has  over- 
taken A,  he  turns  back,  and  reaches  the  starting-point  at 
the  same  instant  that  A  reaches  his  destination.  B  walked 
at  the  rate  of  4  miles  an  hour.  Find  A's  rate,  and  the  dis- 
tance from  the  starting-point  to  where  B  met  A. 

18.  A  tank  can  be  filled  by  three  x^ipes.  A,  B,  and  C, 
when  opened  together,  in  2^^  hours.  If  A  filled  at  the 
same  rate  as  B,  it  would  take  3  hours  for  A,  B,  and  C  to 
fill  the  tank ;  and  the  sum  of  the  times  required  by  A  and 
C  alone  to  fill  the  tank  is  double  the  time  required  by  B 
alone.     In  what  time  can  each  pipe  alone  fill  the  tank? 

19.  The  sum  of  two  numbers  is  4,  and  the  sum  of  their 
fifth  powers  is  244.     Find  the  numbers. 


THEORY  OF  QUADRATIC  EQUATIONS.    261 


XXV.    THEORY  OF  QUADRATIC  EQUA- 
TIONS. 

282.  Sum  and  Product  of  the  Roots. 

Let  ?*i  and  rg  denote  the  roots  of  the  equation  x^  -{-px  =  q. 


By  §  265,  n  =  -P+-iP'  +  ^<i,  and  „  =  -P-^f  +  '^l. 

Adding  these  values,     r^  +  ?'2  =  — — —  =  —P' 
Multiplying  them  together,  we  have 

Hence,  if  a  quadratic  equation  is  in  the  foim  sc^  -{-  px  —  q^ 
the  sum  of  the  roots  is  equal  to  the  coefficient  of  x  with  its  sign 
changed,  and  the  product  of  the  roots  is  equal  to  the  second 
member  with  its  sign  changed. 

1.  Find  the  sum  and  product  of  the  roots  of  the  equation 
2ar^-7a;-15  =  0. 

Transposing  —  16,  and  dividing  by  2,  the  equation  becomes 

2        2 

7  IK 

Hence,  the  sum  of  the  roots  is  -,  and  their  product  —  — • 

2  2 


EXAMPLES. 

Find  by  inspection  the  sum  and  product  of  the  roots  of : 

2.  x^-\-7x  +  6  =  0.  6.   12x'-4:X-\-S  =  0. 

3.  a^-x-\-12  =  0.  7.   9.'B-21.r2  +  7  =  0. 

4.  cf2  +  3a;-l  =  0.  S.   4-x-6a^  =  0. 

5.  Sar^- a- 6  =  0.  9.    Ua:^ +  Sax -\-21a^  =  0. 


262  ALGEBRA. 

283.  Formation  of  Equations. 

By  aid  of  the  principles  of  §  282,  a  quadratic  equation 
may  be  formed  which  shall  have  any  required  roots. 
For,  let  ?*i  and  r^  denote  the  roots  of  the  equation 

x^  -\-px  —  q  =  0.  '  (1) 

Then  by  §  282,  p  =  —  r^  —  r^,^  and  —  g  =  i\t<i. 
Substituting  these  values  in  (1),  we  have 

:>?  —riX  —  r^-{-  TiV^  =  0. 
That  is,  (x  -  r,)  (x  -  r^)  =  0.  (§  93) 

Hence,  any  quadratic  equation  can  be  written  in  the  form 
(x-r,)(x-r,)  =  0,  ^  (2) 

where  Vi  and  rg  are  its  roots. 

Therefore,  to  form  a  quadratic  equation  which  shall  have 
any  required  roots. 

Subtract  each  of  the  roots  from  x,  and  place  the  product  of 
the  resulting  expressions  equal  to  zero. 


and 


1.   Form  the  quadratic  equation  whose  roots  shall  be  4 
7 
4* 

By  the  rule,  (x  -  4)(x +  -\  =  0. 

Multiplying  by  4,  (aj  —  4)  (4  cc  +  7)  =  0. 

Whence,  4  ic2  -  9  a;  -  28  =  0,  Ans. 

EXAMPLES. 

Form  the  quadratic  equations  whose  roots  shall  be : 

2.6,9.  4.1,-1  6.1,  I  8.   -f,0. 

3.  2,-3.         6.   -4,-|.     7.  -|,|.        9.   -|  -|. 


THEORY  OF   QUADRATIC   EQUATIONS.  263 

10.  2a  +  b,a-3b.         12.   ;; -f  7V2,  3  -  7V2. 

11.  a  +  Sm,a-3m.      13.    ^(- Va  +  V^),i(- Va- V6). 

FACTORING. 
284.  Factoring  of  Quadratic  Expressions. 
A  quadratic  expression  is  an  expression  of  the  form 

ax^  -\-  bx  +  c. 

The  principles  of  §  283  serve  to  resolve  sncli  an  expres- 
sion into  two  factors,  each  of  the  first  degree  in  x. 

We  have,      aa^ -h  bx -\- c  =  afa^ -h  —  ^-X  (1) 

\         a      aj 

Now  let  rj  and  r^  denote  the  roots  of  the  equation 

a      a 
By  §  283,  (2),  the  equation  can  be  written  in  the  form 

(x-r{)(x-r2)  =  0. 

Hence,  the  expression  a^-h— +-  can  be  written 

a      a 

(x-ri)(x-r2). 

Substituting  in  (1),  we  have 

ax^  -\-bx-\-c  =  a(x  —  r,)  (x  —  rg). 

hx      c 

But  7\  and  ?*o  are  the  roots  of  the  equation  a^-] 1-  -  =  0, 

a      a 
or  oor^  -f  6aj  -(-  c  =  0 ;  which,  we  observe,  is  obtained  by  placing 
the  given  expression  equal  to  zero. 

We  then  have  the  following  rule : 

To  factor  a  quadratic  expression,  place  it  equal  to  zero,  and 
solve  the  equation  thus  formed. 

Then  the  required  factors  are  the  coefficient  of  o^  in  the  given 
expression,  x  minus  the  first  root,  and  x  minus  the  second  root 


264  ALGEBRA. 

EXAMPLES. 
285.   1.  Factor  Gx'-^-lx-  3. 
Solving  the  equation  6  a:^  +  7  a;  -  3  =  0,  we  have  by  §  265, 


12  12  3  2 

Then  by  the  rule,  6x^-h  1  x  -  5  =  qIx  -^\  Ix  +  -\ 

=  (Sx-  l)(2a;  +  3),  Ans. 
2.   Factor  4  +  13a;-12«2. 

Solving  the  equation  4  +  13  x  —  12  x^  =  0,  we  have  by  §  265, 

^  ^  -  13  -t-  Vl69  +  192  ^  -  13  ±  19  ^      1  ^^4^ 
-24  -24  4       3* 

Whence,  4  + 13  a;- 12x2  =  - 12  (x  +  iVx- 1"! 

=  4(.  +  l)x(-3)(.-|) 

=  (l  +  4ic)(4-3a;),  ^ws. 
Factor  the  following : 

3.  ic2-13a;4-42.  14.  6a^ -23mx +  21m\ 

4.  x'  +  Wx-^-U.  15.  14a^  +  25a;4-6. 

5.  a^-9x-36.  16.  18  ar^  -  15  a;  +  2. 

6.  3a^H-7a;-6.  17.  5-19a:-4a^. 

7.  5a^+18a5  +  16.  18.  18a^  +  31a;  +  6. 

8.  6a^-llaj  +  3.  19.  45  +  7x-12a^. 

9.  15a^-14a:-8.  20.  42  +  23 a; -  10 o^. 

10.  20-7a;-3a^.  21.   24a^-26a;  +  5. 

11.  35-lla;~6«2  22.   80^2^33^.^35 

12.  12  +  28a;-5i«2  23.   21  a^  -  10 a^  -  24 2/1 
.13.   3a^-i7aa;-28al             24.   7 ar^  +  37  a6a;  -  30 a-6l 


THEORY  OF   QUADRATIC    EQUATIONS.  265 

25.    Y^Qtov  2a^-Sxij-2y^-7x  +  4:y  +  6. 
Placing  the  expression  equal  to  zero,  we  have 

2x^-3xy  -2y'^-7x-hiy  +  Q  =  0, 
or  "  2x^-(Pjy  +  7)x  =  2y2-iy  -  0. 

Solving  this  hy  the  formula  of  §  265, 

^_3y  +  7,±V(3y  +  7)-^+16y^-32y-48 


■  ^  3y  +  7:i:V25y2  4-10y+T  _  3y  +  7zb(5y-f  1) 
4  ~  4 

=  -i^or pi- =  22, +  2  or --i^ 

Therefore, 
2x2  -  3a;2/ -  22/2  _  7a;  +  4y  +  6  =  2  [a; -C2y  +  2)]rx -^=^^±-?l 

=  (a;  -  2y  -  2)(2a;  +  y  -  3),  ^ns. 
Factor  the  following : 

26.  a^  +  a;?/-12/  +  7a;  +  72/  +  12. 

27.  a52_a^_22/2  +  a7-52/-2. 

28.  a^-42/^-h3a;  +  10i/-4. 

29.  2»24.7a^_42/2-f a;  +  132/-3. 

30.  3a^-oab-2b^-7a-{-2. 

31.  6-152/-5d;  +  9/4-9a;?/-4a;2 

32.  6a^-dxy  +  xz-Wy^-lSyz-2z'. 

286.  If  the  coefficient  of  x^  is  a  perfect  square,  it  is  con- 
venient to  factor  the  expression  by  the  artifice  of  completing 
the  square  (§  260)  in  connection  with  §  99. 

1.   Factor  9x^-9x-4:. 


By  §  260,  the  expression  9  rc2  -  9  x  will  become  a  perfect  square  by 

q  o 

iing  to  it  the  square  of  — -,  or  —     Then, 
2V'9        2 

9  x2  -  9  X  -  4  =  9  a:2  -  9  X  +  ( §  y  -  2  _  4  Z3  (  3  X -r  -  ]  "^  -  — . 


266  ALGEBRA. 

Factoring  as  in  §  99,  we  have 

9.-9.-4  =  (3.-|  +  |)(.S.x-|-|) 

=  (3x+  l)(3x-4),  Ans. 

If  the  x^  term  is  negative,  the  entire  expression  should  be 
enclosed  in  a  parenthesis  preceded  by  a  —  sign. 

2.  Factor  S-12x-4.x\ 

3-12ic-4x2  =  -(4x2  +  12x-3) 

=  _(4x2+12ic  +  32-9-3) 

=  _  [(2x  +  3)2-12] 

=  (2x  +  3  +  Vl2)x(-  l)(2x  +  3-Vl2) 

=  (2V3  +  3  +  2 X) (2\/3  -  3  -  2 x),  ^/is. 

EXAMPLES. 
Factor  the  following :  ' 

3.  ic2-5a;  +  4.  9.  36  x" -{- 24:  x  -  5. 

4.  4a^  +  16x  +  15.  10.  4.x'  +  5x-6. 

5.  9a;2_l8x  +  8.  11.  25x'-\-30x  +  6. 

6.  16ar2+16a;-21.  12.  4  +  12a)-9a^. 

7.  a^  +  2a;-ll.  13.  49 a^^  +  56 ic  + 12. 

8.  4x2  +  4a^-l.  14.  5-\-SSx-16a^. 

287.  Certain  trinomials  of  the  form  ax*  +  fta?^  +  c,  where 
a  and  c  are  perfect  squares,  may  be  resolved  into  two  fac- 
tors by  the  artifice  of  completing  the  square. 

1.   Factor  9x^-28x^-^4:. 

By  §  96,  the  expression  will  become  a  perfect  square  if  its  middle 
term  is  —  12  x^. 

Thus,    9ic4- 28x2 +  4  =(9^4 -12^2 +  4) -16x2 

^(3x2-2)2-(4x)2 

=  (3x2-2 +  4x)(3x2-2-4x)  (§99) 

=  (3x2  +  4x-2)(3x2-4x-2),  Ans. 


THEORY   OF   QUADRATIC   EQUATIONS.  267 

2.  Factor  a'  +  aV/  +  b\ 

=  (a2+&2)2_a-2^2 

=  (a2  +  62  +  «6)(a2  4.?,-2_a6) 

=  (a2  +  a6  +  62)  (a2  _  a&  +  6'0»  ^««- 

3.  Factor  a;^  +  1. 

X4  +  1  =(iC*  +  2ic2  +  1)_  2x2 

=  (a;2  +  l)2_(a;\^)2 

=  (aj2  +  a;  v/2  +  1) (x2  _  a^  V2  +  1),  ^ns. 

EXAMPLES. 
Factor  the  following : 

4.  x'-\-2a^  +  9.  12.  a;*  +  16. 

5.  a;^-19a;2-h25.  13.  x'-5a^  +  l. 

6.  4a<  +  7a262-|-166^  14.  9 a* -  55 a^x^  _j_ 25 aj*. 

7.  9a;^-28ic2/  +  4y.  15.  16  a*  +  47  aV  ^.  36  m*. 

8.  16m^-mV4-n'.  16.  25  a^^  -  21  «2  +  4. 

9.  4a^-53a2  +  49.  17.  25 m^  +  36 mV  +  16 a.-*. 

10.  9a;*  +  5ar^  +  9.  18.    16 a^  -  60 a^^y.  _^  49 ^ 

11.  4m^-13m2  +  4.  19.   36  a^  -  68  a^ft^  +  25  6^ 


Certain  equations  of  the  fourth  degree  may  be 
solved  by  factoring  the  first  member  by  the  method  of 
§  287,  and  then  proceeding  as  in  §  267. 

1.    Solve  the  equation  a;*  +  1  =  0. 

By  Ex.  3,  §  287,  the  equation  may  be  written 

(x2  +  X  V2  +  l)(a;2  -  X  V2  +  1)=  0. 
Then,  as  in  §  267,  a:2  +  x  V2  +  1  =  0,  and  x2  -  x  \/2  +  1  =  0. 
Solving  the  equation  a;2  +  x  V'2  4- 1  =  0,  we  have  by  §  265, 


_  V2  ±  V2^^      -  V2  ± 

X  = = =  =^ 

2  2 


268  ALGEBRA. 

Solving  the  equation  aj^  —  x  V2  +  1  =  0,  we  have 


EXAMPLES. 

Solve  the  following : 

3.  a?4-18a;2H-9  =  0.  6.   .t^  -  Ooj^  +  9  =  0. 

4.  4aj^-5aj24-l  =  0.  7.   o.-^  +  81  =  0. 


DISCUSSION  OF  THE  GENERAL  EQUATION. 
By  §  265,  the  roots  of  the  equation  x^  -\- px  =  q  are 


We  will  now  discuss  these  values  for  all  possible  real 
values  of  p  and  q. 

I.  Suppose  q  positive. 

Since  p^  is  essentially  positive  (§  186),  the  expression 
under  the  radical  sign  is  positive,  and  greater  than  p^. 

Therefore,  the  radical  is  numerically  greater  than  p. 

Hence,  r^  is  positive,  and  rg  is  negative. 

If  p  is  positive,  rg  is  numerically  greater  than  Vi ;  that  is, 
the  negative  root  is  numerically  the  greater. 

If  j9  is  zero,  the  roots  are  numerically  equal. 

If  p  is  negative,  7\  is  numerically  greater  than  rgj  that 
is,  the  positive  root  is  numerically  the  greater. 

II.  Suppose  q  =  0. 

The  expression  under  the  radical  sign  is  now  equal  to  p^. 
Therefore,  the  radical  is  numerically  equal  to  p. 
If  p  is  positive,  r^  is  zero,  and  rg  is  negative. 
If  p  is  negative,  Vi  is  positive,  and  rg  is  zero. 


THEORY  OF   QUADRATIC   EQUATIONS.         269 

III.  Suppose  q  negative,  and  4g  niimericaUy  <p^. 

The  expression  under  the  radical  sign  is  now  positive,  and 
less  than  p'^. 

Therefore,  the  radical  is  numerically  less  than  p. 
If  p  is  positive,  both  roots  are  negative. 
If  p  is  negative,  both  roots  are  positive. 

IV.  Suppose  q  negative,  and  4  q  numerically  equal  to  p'. 
The  expression  under  the  radical  sign  now  equals  zero. 
Hence,  Vi  is  equal  to  i^. 

If  p  is  positive,  both  roots  are  negative. 
If  p)  is  negative,  both  roots  are  positive. 

V.  Suppose  q  negative,  and  4  q  numerically  >  p^. 

The  expression  under  the  radical  sign  is  uow  negative. 
Hence,  both  roots  are  imaginary  (§  248). 

The  roots  are  both  rational  or  both  irrational,  according 
as  p'^  -\-  ^q  is  or  is  not  a  perfect  square. 

EXAMPLES. 

290.   1.    Determine  by  inspection  the  nature  of  the  roots 
of  the  equation  2  ic^  —  5  a:  —  18  =  0. 

The  equation  may  be  written  x^ -  =  ^;  here  p  =  —  -  and  g  =  9. 

Since  q  is  positive  and  p  negative,  the  roots  are  one  positive  and 
the  other  negative  ;  and  the  positive  root  is  numerically  the  greater. 

In  this  case,  p'^  +  ^  q  =  ^  Jf  ZQ  =  — ;  a  perfect  square. 

4  4 

Hence,  the  roots  are  both  rational. 

Determine  by  inspection  the  nature  of  the  roots  of  the 
following : 

2.  6a52  +  7a;-5  =  0.  7.  l&a^-^  =  0. 

3.  10a^  +  17a;  +  3  =  0.  8.  ^:x?-l  =  12x. 

4.  4ar^-iB  =  0.  9.  2bx' -\-^0x +  ^  =  0. 

5.  4a;2_20.^.-f  25  =  0.  10.  7.^•2  +  3a;  =  0. 

6.  ar^- 21  a; +  200  =  0.  11.  41  a;  =  20  ar^  +  20. 


270  ALGEBRA. 

XXVI.     ZERO  AND  INFINITY. 

VARIABLES  AND  LIMITS. 

291.  A  variable  quantity,  or  simply  a  variable,  is  a  quan- 
tity which  may  assume,  under  the  conditions  imposed  upon 
it,  an  indefinitely  great  number  of  different  values. 

A  constant  is  a  quantity  which  remains  unchanged 
throughout  the  same  discussion. 

292.  A  limit  of  a  variable  is  a  constant  quantity,  the  dif- 
ference between  which  and  the  variable  may  be  made  less 
than  any  assigned  quantity,  however  small,  but  cannot  be 
made  equal  to  zero. 

In  other  words,  a  limit  of  a  variable  is  a  fixed  quantity 
to  which  the  variable  approaches  indefinitely  near,  but 
never  actually  reaches. 

Suppose,  for  example,  that  a  point  moves  from  A  towards 
B  under  the  condition  that  it  shall  move,  during  succes- 
sive equal  intervals  of  time, 

first  from  A  to   C,   half-way   f f  f    f     T 

between   A   and   B;   then   to 

D,  half-way  between  C  and  B ;  then  to  E,  half-way  between 

D  and  B;  and  so  on  indefinitely. 

In  this  case,  the  distance  between  the  moving  point  and 
B  can  be  made  less  than  any  assigned  quantity,  however 
small,  but  cannot  be  made  equal  to  zero. 

Hence,  the  distance  from  A  to  the  moving  point  is  a  vari- 
able which  approaches  the  constant  value  AB  as  a  limit. 

Again,  the  distance  from  the  moving  point  to  5  is  a 
variable  which  approaches  the  limit  0. 

293.  A  problem  is  said  to  be  indeterminate  when  the 
number  of  solutions  is  indefinitely  great.     (Compare  §  159.) 


ZERO   AND  INFINITY.  271 

294.  Interpretation  of  ^• 

Consider  the  series  of  fractions 

3'  .3'  .03'  .003'"*' 
where  each  denominator  after  the  first  is  one-tenth  of  the 
preceding  denominator. 

It  is  evident  that,  by  sufficiently  continuing  the  series, 
the  denominator  may  be  made  less  than  any  assigned  quan- 
tity, however  small,  and  the  value  of  the  fraction  greai^^er 
than  any  assigned  quantity,  however  great. 

In  other  words. 

If  the  numerator  of  a  fraction  remains  constant^  icliile  the 
denominator  approaches  the  limit  0,  the  value  of  the  fraction 
increases  without  limit. 

It  is  customary  to  express  this  principle  as  follows : 
a 

-  =  QO. 
<) 

Note.     The  symbol  oo  is  called  Infinity. 

295.  Interpretation  of  —  • 

Consider  the  series  of  fractions 
a    a      a        a 
3'  30'  300'  3000''"' 
where  each  denominator  after  the  first  is  ten  times  the  pre- 
ceding denominator. 

It  is  evident  that,  by  sufficiently  continuing  the  series, 
the  denominator  may  be  made  greater  than  any  assigned 
quantity,  however  great,  and  the  value  of  the  fraction  less 
than  any  assigned  quantity,  however  small. 

In  other  words, 

If  the  numerator  of  a  fraction  remains  constant,  while  the 
denominator  increases  without  limit,  the  value  of  the  fraction 
approaches  the  limit  0. 


272  ALGEBRA. 

It  is  customary  to  express  this  principle  as  follows; 

00 

296.  It  must  be  clearly  understood  that  no  literal  meaning 
can  be  attached  to  such  results  as 

-=500.  or  —  =  0:r 

for  there  can  be  no  such  thing  as  division  unless  the  divisor 
is  a  finite  quantity. 

If  such  forms  occur  in  mathematical  investigations,  they 
must  be  interpreted  as  indicated  in  §§  294  and  295.  (Com- 
pare note  to  §  395.) 

THE  PROBLEM  OF  THE  COURIERS. 

297.  The  discussion  of  the  following  problem  will  serve 
to  further  illustrate  the  form  -,  besides  furnishing  an  inter- 

0 

pretation  of  the  form  -• 

The  Problem  of  the  Couriers.  Two  couriers,  A  and  B, 
are  travelling  along  the  same  road  in  the  same  direction, 
RR\  at  the  rates  of  m  and  n  miles  an  hour,  respectively. 
If  at  any  time,  say  12  o'clock,  A  is  at  P,  and  B  is  a  miles 
beyond  him  at  Q,  after  how  many  hours,  and  how  many 
miles  beyond  P,  are  they  together  ? 

R  p  q  BT 

I I \ ^ I 


Let  A  and  B  meet  x  hours  after  12  o'clock,  and  y  miles 

beyond  P. 

They  will  then  meet  y  —  o.  miles  beyond  Q. 

Since  A  travels  mx  miles,  and  B  nx  miles,  in  x  hours,  we 

have 

(        y  =  mx. 

Xy  —  a  =  nx. 


ZERO   AND   INFINITY.  273 

Solving  these  equations,  we  obtain 

a  1  am 

X  = ,  and  y  = 

m  —  n  m  —  n 

We  will  now  discuss  these  results  under  different  hypoth- 
eses. 

1.  m  >  n. 

In  this  case,  the  values  of  x  and  y  are  positive. 

Hence,  the  couriers  will  meet  at  some  time  after  12 
o'clock,  and  at  some  point  to  the  rigid  of  P. 

This  corresponds  with  the  hypothesis  made ;  for  if  m  is 
greater  than  n,  A  is  travelling  faster  than  B ;  and  it  is  evi- 
dent that  he  will  eventually  overtake  him  at  some  point 
beyond  their  positions  at  12  o'clock. 

2.  tn  <  n. 

In  this  case,  the  values  of  x  and  y  are  negative. 

Hence,  the  couriers  met  at  some  time  before  12  o'clock, 
and  at  some  point  to  the  left  of  P.     (Compare  §  10.) 

This  corresponds  with  the  hypothesis  made ;  for  if  m  is 
less  than  7i,  A  is  travelling  more  slowly  than  B ;  and  it  is 
evident  that  they  must  have  been  together  before  12  o'clock, 
and  before  they  could  have  advanced  as  far  as  P. 

3.  m  =  ?i,  or  m  —  nz=  0. 

In  this  case,  the  values  of  x  and  y  take  the  forms  -  and 

0 
— ,  respectively.  • 

If  m  —  n  approaches  the  limit  0,  x  and  y  increase  with- 
out limit  (§  294) ;  hence,  if  m  =  ii,  no  finite  values  can  be 
assigned  to  x  and  y,  and  the  problem  is  impossible. 

Thus,  a  result  in  the  form  -  indicates  that  the  problem  is 


This  interpretation  corresponds  with  the  hypothesis  made ; 
for  if  m  =  n,  the  couriers  are  a  miles  apart  at  12  o'clock, 
and  are  travelling  at  the  same  rate  ;  and  it  is  evident  that 
they  never  could  have  be§B^_and  never  will  be  together. 


CALIFO^J 


274  ALGEBRA. 

4.  a  =  0,  and  m  >  n  or  m  <  n. 

In  this  case,  x  =  0  and  2/  =  0. 

Hence,  the  couriers  are  together  at  12  o'clock,  at  the 
point  P. 

This  corresponds  with  the  hypothesis  made ;  for  if  a  =  0, 
and  m  and  n  are  unequal,  the  couriers  are  together  at  12 
o'clock,  and  are  travelling  at  unequal  rates ;  and  it  is  evi- 
dent that  they  never  could  have  been  together  before  12 
o'clock,  and  never  will  be  together  afterwards. 

5.  a  =  0,  and  m  =  n. 

In  this  case,  the  values  of  x  and  y  take  the  form  — 

If  a  =  0,  and  m  =  n,  the  couriers  are  together  at  12  o'clock, 
and  are  travelling  at  the  same  rate. 

Hence,  they  always  have  been,  and  always  will  be  together. 

In  this  case  the  number  of  solutions  is  indefinitely  great ; 
for  any  value  of  x  whatever,  together  with  the  correspond- 
ing value  of  y,  will  satisfy  the  given  conditions. 

Thus,  a  result  in  the  form  -  indicates  that  the  problem  is 
indeterminate  (§  293). 


NDETERMINATE   EQUATIONS.  275 


XXVII.    INDETERMINATE  EQUATIONS. 

298.  It  was  shown  in  §  159  that  a  single  equation  con- 
taining two  or  more  unknown  quantities  is  satisfied  by  an 
indefinitely  great  number  of  sets  of  values  of  these  quanti- 
ties. If,  however,  the  unknown  quantities  are  required  to 
satisfy  other  conditions,  the  number  of  solutions  may  be 
finite. 

We  shall  consider  in  the  present  chapter  the  solution  of 
indeterminate  equations  of  the  first  degree,  containing  two 
or  more  unknown  quantities,  in  which  the  unknown  quanti- 
ties are  restricted  to  positive  integral  values. 

299.  Solution  of  Indeterminate  Equations  in  Positive 
Integers. 

1.   Solve  the  equation  7x-\-5y  =  118  in  positive  integers. 

Dividing  by  5,  the  smaller  of  the  two  coefficients,  we  have 

5  5 

Or,  lEp3  =  2S-x-y. 

5 

Since,  by  the  conditions  of  the  problem,  x  and  y  must  be  positive 

2  X  —  3 

integers,  it  follows  that  must  be  an  integer. 

5 

Let  this  integer  be  represented  by  p. 

Then,  ^^f-^  =p,  or  2  X  -  3  =  5i).  (1) 

5 

Dividing  (1)  by  2,      x  -  1  -  i  =  2j9  +  ^. 
-,.  -  2  2 

Or,  x-l-2p=P^- 

Since  x  and p  are  integers,  x-l-2p  is  also  an  integer ;  and  there- 
fore ^  must  be  an  integer. 
2 

Let  this  integer  be  represented  by  q. 


276  ALGEBRA. 

Then,  ^-i-i  =  q,  or  p  =  2q-l. 

Substituting  in  (1),        2 x  -  3  =  10 g  -  5. 

Whence,  2ic  =  lOg^  -  2,  and  x  =  5g  -  1.         (2) 

Substituting  this  value  in  the  given  equation, 

35g-7  +  5?/=118. 
Whence,  5  y  =  125  -  35  g,  and  y  =  25  -  7  g.     (3) 

Equations  (2)  and  (3)  form  what  is  called  the  general  solution  in 
integers  of  the  given  equation. 

Now  if  q  is  zero,  or  any  negative  integer,  x  will  be  negative  ;  and  if 
q  is  any  positive  integer  greater  than  3,  y  will  be  negative. 

Hence,  the  only  positive  integral  values  of  x  and  y  which  satisfy  the 
given  equation  are  those  arising  from  the  values  1,  2,  3  of  g-. 

If  g  =  1,  5C  =  4,  and  y  =  18;  if  g  =  2,  x  =  9,  and  y  =  ll;  if  g  =  3, 
X  =  14,  and  y  =  4. 

2.  In  how  many  ways  can  the  sum  of  $  15  be  paid  with 
dollars,  half-dollars,  and  dimes,  the  number  of  dimes  being 
equal  to  the  number  of  dollars  and  half-dollars  together  ? 


JiGt                                     X  =  the  number  of  dollars, 

y  =  the  number  of  half-dollars, 

and                                         z  =  the  number  of  dimes. 

Then,            lOx  +  by  +  z  =  160, 

(1) 

and                                         z  =  x  +  y. 

(2) 

Subtracting  (2)  from  (1), 

10a;  +  5?/  =  150-x-?/,  or  llx-f6?/  = 

:150.     (3) 

Dividing  by  6,  x -\- — -{■  y  =  25. 

5x 
Then,  — -  must  be  an  integer  ;  or,  x  must  be  a  multiple  of  6. 
6 

Let  X  =  6p,  where  p  is  an  integer. 

Substituting  in  (3),  dSp  +  6y  =  150,  or  y  =  25-  Up. 

Substituting  in  (2),  z  =  6p -\- 26  —  Up  =26  -  dp. 

The  only  positive  integral  solutions  are  when  p  =  1  or  2;  ifp  =  l, 
X  =  6,  y=U,  and  5;  =  20  ;  if  p  =  2,  x  =  12,  y  =  3,  and  z  =  15. 

Then  the  number  of  ways  is  two  ;  either  6  dollars,  14  half-dollars, 
and  20  dimes ;  or  12  dollars,  3  half-dollars,  and  15  dimes. 


INDETERMINATE   EQUATIONS.  277 

EXAMPLES. 
Solve  the  following  in  positive  integers : 

3.  2x-^3y  =  21.  9.   43 a; -|- 10 y  =  719. 

4.  7x-\-4:y  =  S0.  10.   Sx-\-19y  =  700. 


5.  7a; +  382/ =  211. 

6.  31  a; +  9  2/ =  1222. 


11. 


2x-^Sy  —  oz  =  -S. 
5x-y-\-4:Z  =  21. 


7.  24a;  +  72/  =  422.  (  Sx-2y -z  =  -^  57. 

8.  8a;-f  67  2/  =  158.  ^^-   |  6a;-f- lly  +  22  =  348. 

Solve  the  following  in  least  positive  integers : 

13.  4.x-3y  =  5.  16.   21a;-8y  =  -25. 

14.  5x-7y  =  ll.  17.   13a;  -  30.1/ =  61. 
15.^9  a; -4  2/ =  128.  18.    17x-5Sy  =  -79. 

19.  In  how  many  different  ways  can  the  sum  of  ^  2.10  be 
paid  with  twenty-five  and  twenty-cent  pieces  ? 

20.  In  how  many  different  ways  can  the  sum  of  f  3.90  be 
paid  with  fifty  and  twenty-cent  pieces  ? 

21.  Find  two  fractions  whose  denominators  are  9  and  5, 
respectively,  and  whose  sum  .shall  be  equal  to  -y^'^. 

^22.   In  how  many  different  Ways  can  the  sum  of  $  5.10  be 

paid  with  half-dollars,  quarter-dollars,  and  dimes,  so  that 
the  whole  number  of  coins  used  shall  be  20  ? 

23.  A  farmer  purchased  a  certain  number  of  pigs,  sheep, 
and  calves  for  $  160.  The  pigs  cost  $  3  each,  the  sheep  f  4 
each,  and  the  calves  ^  7  each ;  and  the  number  of  calves 
was  equal  to  the  number  of  pigs  and  sheep  together.  How 
many  of  each  did  he  buy  ? 

24.  In  how  many  different  ways  can  the  sum  of  $5.45 
be  paid  with  quarter-dollars,  twenty-cent  pieces,  and  dimes, 
so  that  twice  the  number  of  quarters  plus  5  times  the  num- 
ber of  twenty-cent  pieces  shall  exceed  the  number  of  dimes 
by  36? 


^ 


278  ALGEBRA. 


XXVIIL    RATIO  AND  PROPORTION. 

300.  The  Ratio  of  one  number  to  another  is  the  quotient 
obtained  by  dividing  the  first  number  by  the  second. 

Thus,  the  ratio  of  a  to  6  is  - ;  and  it  is  also  expressed  a :  b. 

h 

301.  A  Proportion  is  a  statement  that  two  ratios  are 
equal. 

The  statement  that  the  ratio  of  a  to  6  is  equal  to  the 
ratio  of  c  to  d,  may  be  written  in  either  of  the  forms 

,  T        a     c 

a:b  =  c:  d,  01  -  =  -' 

b      d  ■       ' 

302.  The  first  and  fourth  terms  of  a  proportion  are  called 
the  extremes,  and  the  second  and  third  terms  the  means. 

The  first  and  third  terms  are  called  the  antecedents,  and 
the  second  and  fourth  terms  the  consequeyits. 

Thus,  in  the  proportion  a\b  =  c:d,  a  and  d  are  the  ex- 
tremes, b  and  c  the  means,  a  and  c  the  antecedents,  and 
b  and  d  the  consequents. 

303.  If  the  means  of  a  proportion  are  equal,  either  mean 
is  called  a  Mean  Proportional  between  the  first  and  last 
terms,  and  the  last  term  is  called  a  Third  Proportional  to 
the  first  and  second  terms. 

Thus,  in  the  proportion  a:b  =  b:  c,  b  is  a  mean  propor- 
tional between  a  and  c,  and  c  is  a  third  proportional  to 
a  and  6. 

304.  A  Fourth  Proportional  to  three  quantities  is  the 
fourth  term  of  a  proportion  whose  first  three  terms  are  the 
three  quantities  taken  in  their  order. 


RATIO   AND   PROPORTION.  279 

Thus,  in  the  proportion  a:  b  =  c:  d,  d  is  a  fourth  propor- 
tional to  a,  h,  and  c. 

305.  A  Continued  Proportion  is  a  series  of  equal  ratios, 
in  which  each  consequent  is  the  same  as  the  following  ante- 
cedent; as, 

a\h  =  h  :c  =  c:d  =  d'.e. 


PROPERTIES  OF  PROPORTIONS. 

>J 

306.  In  any  proportion^  the  product  of  the  extremes  is  equal 

to  the  ])rodu(it  of  the  means. 

Let  the  proportion  be     a  :  6  =  c  :  rf. 

Then  by  §301,  ^=L 

0      d 

Clearing  of  fractions,        ad  =  he.  , 

307.  A  mean  proportional  between  two  quantities  is  equal 
to  the  square  root  of  their  product. 

Let  the  proportion  be     a:b  =  b:c. 

Then,  b^  =  ac.  (§  306) 

Whence,  b  =  Vac. 

308.  From  the  equation  ad  —  be,  we  obtain 

a  =  — ,  and  6=  — 
d  c 

That  is,  in  any  proportion,  either  extreme  is  equal  to  the 
product  of  the  means  divided  by  the  other  extreme;  and 
either  mean  is  equal  to  the  product  of  the  extremes  divided 
by  the  other  mean. 

A  309.  (Converse  of  §  306.)  If  the  product  of  two  quantities 
is  equal  to  the  ptroduct  of  two  others,  one  pair  may  be  made 
the  extremes,  and  the  other  pair  the  means,  of  a  proportion. 


280  ALGEBRA. 

Let  ad  =  be. 

T-k-   -J-      I.    X.  7  ad     be         a     c 

Dividing  by  bd,  f:i  =  lZi^  ^^'  i.  =  y 

bd     bd         b     d 

Whence  by  §  301,  a:b  =  e:d. 

In  like  manner,  we  may  prove  that 

a:  e=:b:  dj 

e:d  =  a:b,  etc. 

310.  In  any  proportion,  the  terms  are  in  proportion  by 
Alternation;  that  is,  the  first  term  is  to  the  third  as  the  second 
term  is  to  the  fourth. 

Let  the  proportion  be   a:  b  =  c:  d. 

Then,  ad  =  be.  (§  306) 

Whence,  a:c  =  b:d.  (§  309) 

311.  In  any  proportion,  the  terms  are  in  proportioyi  by 
Inversion ;  that  is,  the  second  term  is  to  the  first  as  the  fourth 
temn  is  to  the  third. 

Let  the  proportion  be   a  :  6  =  c  :  d. 

Then,  ad  =  be.  (§  306) 

Whence,  b:a  =  d'.c.  (§  309) 

V  312.  In  any  proportion,  the  terms  are  in  proportion  by 
Composition ;  that  is,  the  sum  of  the  first  two  terms  is  to  the 
first  term  as  the  sum  of  the  last  two  terms  is  to  the  third  term. 

Let  the  proportion  be   a:b  =  c:d. 

Then,  ad  =  be. 

Adding  each  member  of  the  equation  to  ac, 
ac -\-  ad  =^  ac  ■\-  be. 

Or,  a{c  +  cZ)  =  c{a  -f  b). 

Whence,  a  ^  6  :  a  =  c  +  c« :  c.  (§  309) 

In  like  manner,  we  may  prove  that 

a  +  b  :  b  =  c  -\-  d  :  d. 


RATIO   AND   PROPORTION.  281 

'313.  In  any  proportion,  the  terms  are  in  i)roportion  by 
Division;  that  is,  the  difference  of  the  first  two  terms  is  to 
the  first  term  as  the  difference  of  the  last  two  terms  is  to  the 
third  term,. 

Let  the  proportion  be   a  :  6  =  c  :  d. 

Then,  ad  =  he. 

Subtracting  each  member  of  the  equation  from  ac, 
ac  —  ad  =  ac  —  he. 

Or,  a{c  —  d)  —  c{a  —  h). 

Whence,  a  —  b:a  =  c  —  d:c. 

Similarly,  a  —  h  :  b  =  c  —  d  :  d. 

V  314.  In  any  iiroportion^  the  terms  are  in  proportion  hy 
Composition  and  Division ;  that  is,  the  sum  of  the  first  two 
terms  is  to  their  difference  as  the  sum  of  the  last  two  terms 
is  to  their  difference. 


(1) 
(2) 


^315.  In  a  series  of  equal  ratios,  any  antecedent  is  to  its  con- 
sequent as  the  sum  of  all  the  antecedents  is  to  the  sum  of  all 
the  consequents. 

Let  a:b  =  c:d  =  e:f. 

Then  by  §  306,  ad  =  he, 

and  af=  be.  u^ 

Also,  ah  =  6g. —      ( 

Adding,     a(h -\- d -\-f)  =  b(a -{- c  +  e). 

Whence,  a:6  =  a  +  c  +  e:6  +  d-f/.         (§309) 


Let  the  proportion  be 

a:b  =  c:d. 

Then  by  §  312, 

a  +  b     c-\-d 
a            c 

And  by  §313, 

a  —  b      c  —  d 
a            c 

Dividing  (1)  by  (2), 

a-\-b      c-^d 
a  —  h     c  —  d 

Whence,                a-\-h 

:  a  —  b  =  c  -\-  d  :  c 

282  ALGEBRA. 


^ 


a  _ 

_c 

b~ 

~d 

ma 
mb 

_nc 

J 


In  like  manner,  the  theorem  may  be  proved  for  any  num- 
ber of  equal  ratios. 

316.  In  any  proportion,  if  the  first  two  terms  be  multiplied 
by  any  quantity,  as  also  the  last  two,  the  resulting  quantities 
loill  be  in  proportion. 

Let  the  proportion  be      a:b  =  c:  d. 
Then, 

Therefore, 

Whence,  ma  :  mb  —  nc  :  nd. 

In  like  manner,  we  may  prove  that 

a     b  _c    d 

m  '  m     n  '  n 

Note.  Either  m  or  71  may  be  unity  ;  that  is,  eitlier  couplet  may 
be  multiplied  or  divided  without  multiplying  or  dividing  the  other. 

0 

317.  In  any  proportion,  if  the  first  and  third  terms  be  mul- 
tiplied by  any  quantity,  as  also  the  second  and  fourth  terms, 
the  resulting  quantities  ivill  be  in  proportion. 

Let  the  proportion  be      a:  b  =  c:  d. 
Then, 

Therefore, 

Whence,  ma  :  71b  =  mc  :  nd. 

In  like  manner,  we  may  prove  that 

a     b  _  c    d 
m'  n     m'  n 

Note.     Either  m  or  w  may  be  unity. 


a  _ 
V 

_  c 
~d 

ma 
nb 

mc 
nd 

RATIO    AND   PROPORTION.  283 

318.  In  any  number  of  j^roportions,  the  products  of  the  cor- 
responding terms  are  in  proportion. 

Let  the  proportions  hQ  a:h  =  c:d, 
and  e:f=g:h. 

Then,  ^  =  £,and«  =  f. 

b     d  f     h 

Multiplying  these  equals,  we  have 

a      e      c      q         ae      cq 
_  y  ..  —  _  V  —    or  =  — ^« 

b^f~d^h'       bf     dh 
Whence,  ae:bf—cg:  dh. 

In  like  manner,  the  theorem  may  be  proved  for  any  num- 
ber of  proportions. 

319.  In  any  propoi'tion,  like  powers  or  like  roots  of  the 
terms  are  in  proposition. 

Let  the  proportion  he  a:b  =  c:  d. 

1=1- ■ 

Therefore,  ^  =  ^. 

Whence,  a"" -.  b""  =  c"" :  d\ 

In  like  manner,  we  may  prove  that 

y/a  :  -Vb  =  -Vc  :  -^d. 

320.  If  three  quantities  are  in  continued  proportion,  the 
first  is  to  the  third  as  the  square  of  the  first  is  to  the  square 
of  the  seco7id. 

Let  a:b  =  b:c. 

Then,  «  =  *. 

b      c 

Therefore,  ?  x  ^  =  ?  X  |,  or  ?5  =  ^'. 

b      c     b      b         c     b^ 

Whence,  a:c  =  a^:bK 


284  ALGEBRA. 

321.  If  four  quantities  are  in  continued  proxiortion,  the 
first  is  to  the  fourth  as  the  mihe  of  the  first  is  to  the  cube  of 
the  second. 

Let  a:  b  =  b  :  c  =  c:  d. 

a_6_c 
b      c^    d 


Then, 


rpi        o  a     b      c      a     a     a         a     a^ 

Therefore,  -  x  -  X     =  -  x  -  x  -,  or  -  =  -• 

b      c     d      b      b      b         d      b^ 

Whence,  a  :  d  =  a^ :  b^. 


PROBLEMS. 
322.   1.  Solve  the  equation 

2x+3:2x-3  =  2b-\-a:2b-a. 

By  §314,  ix:6  =  4h  :2a. 

Dividmg  the  first  and  third  terms  by  4,  and  the  second  and  fourth 
terms  by  2  (§  317),  we  have 

X  :  'S  =  b  :  a. 

Whence  by  §  308,  x  =  —,  A7is. 

a 

2.  li  x:y  =(x-^z)- :  (y  -{-  %f^  prove  that  ^  is  a  mean  pro- 
portional between  x  and  y. 

From  the  given  proportion,  y{x  +  zY  =  x(y  +  zY-  (§  306) 

Or,  .  x^i/  +  2  xyz  +  yz"^  =  xy^  -\- 2  xyz  +  xz^. 

Or,  x^y  —  xy"^  =  xz^  —  yz^.  ^\:^ 

Dividing  hy  x  —  y,  xy  =  z^.       ,    ^    "> 

Therefore,  2  is  a  mean  proportional  between  x  and  y  (§  307). 

3.  If  -  =  -,  prove  that 

b     d 

a'  -  b'  :  a'  -  Sab  =  c^  -  d' :  c"  -  3cd. 
Let  -=-  =  x:  whence,  a  =  bx.  "J   I 

--1 

a^-b^         &2x2  -b^    _  x"^-!  _  d^ c^  -  d^       - 

-^*^®^'  a2-3a6~6%2_362a;-x2-3x~c^_3c~c2-3c(^*       ^ 

d^      d 
Whence,      a^  -  b^  :  a"^  -  Z  ab  =  c^  -  d^  :  c2  -  3  cd. 


RATIO   AND   PROPORTION.  285 

4.  Find  a  fourth  proportional  to  35,  20,  and  14. 

5.  Find  a  mean  proportional  between  18  and  50. 

6.  Find  a  third  proportional  to  ^  and  ^. 

7.  Find  the  second  term  of  a  proportion  whose  first, 
third,  and  fourth  terms  are  5^,  4J,  and  1|. 

8.  Find  a  third  proportional  to  a^  —  9  and  a  —  3. 

^'     9.   Find  a  mean  proportional  between  o^  and  18^. 
\j  10.   Find  a  mean  proportional  between 

; —  and  — — — 

X  +  4:  x  +  2 

Solve  the  following  equations :  h 

^   11.   5a;-3a:5a;  +  3a  =  7a-5:13a-5.  I 

\i  12.   2a;-l:3a;-l  =  7a;  +  l:5a;-3.  "" 

13.   .x-^  -  16  :  a^  -  25  =  ar^  -  2a;  -  24  :  .-cf  -  3a;  -  10. 

"^    14.    1-Vr^:l  +  VT=^=V6-V&^:  V6+V6^. 


15. 


ax  —  by:bx-^ay  =  a^  —  b^:2ab. 
xy  =  a^b^. 

)      16.    Find  two  numbers  in  the  ratio  16  to  9  such  that,  if 
each  be  diminished  by  8,  they  shall  be  in  the  ratio  12 :  5. 

17.  Divide  36  into  two  parts  such  that  the  greater  dimin- 
ished by  4  shall  be  to  the  less  increased  by  3  as  3  is  to  2. 

18.  Find  two  numbers  such  that,  if  4  be  added  to  each, 
they  will  be  in  the  ratio  5  to  3 ;  and  if  11  be  subtracted 
from  each,  they  will  be  in  the  ratio  10  to  3. 

19.  There  are  two  numbers  in  the  ratio  3  to  4,  such  that 
their  sum  is  to  the  sum  of  their  squares  as  7  is  to  50.  What 
are  the  numbers  ? 

M       20.    If  7a;-42:8a;-32  =  4?/-72:3y-82,  provethat 
2  is  a  mean  proportional  between  x  and  y. 


286  ALGEBRA. 

21.  If  ma  -\- nb  :  pa -^  qb  =  mb  -}- nc  :  pb  -{-  qc,  prove  that 
6  is  a  mean  proportional  between  a  and  c. 

22.  If  2a-6:4a  +  36  =  2c-(^:4c-f 3d,  prove  that 
a:b  =  c\d. 

23.  If  8  cows  and  5  oxen  cost  four-fifths  as  much  as  9 
cows  and  7  oxen,  what  is  the  ratio  of  the  price  of  a  cow  to 
that  of  an  ox  ? 

24.  Given  (a^  +  ab)x  +  (b^  -  ab)y=  (a"  +  b^)x  -  (a?  -  b^)y ; 
find  the  ratio  of  x  to  y. 

25.  Find  a  number  such  that  if  it  be  added  to  each  term 
of  the  ratio  5  :  3,  the  result  is  f  of  what  it  would  have  been 
if  the  same  number  had  been  subtracted  from  each  term. 

If  -  =  -,  prove  that 
b     d!  ^ 

26.  2a  +  36:2a-36  =  2c  +  3d:2c-3d 

27.  a^  +  2  a6  :  3  a&  -  4  62  =  c2  +  2  c(^  :  3  c(i  -  4  dl 

28.  a«  -  a'b  +  ab^  :a^-W  =  &  -  cH  -f  cd^ :  c^  -  d\ 

29.  The  population  of  a  town  increased  2.6  per  cent  from 
1870  to  1880.  The  number  of  males  decreased  3.8  per  cent 
during  the  same  period,  and  the  number  of  females  increased 
10.6  per  cent.     Find  the  ratio  of  males  to  females  in  1870. 

30.  Each  of  two  vessels  contains  a  mixture  of  wine  and 
water ;  in  one  the  wine  is  to  the  water  as  1  to  3,  and  in  the 
other  the  wine  is  to  the  water  as  3  to  5.  A  mixture  from 
the  two  vessels  is  composed  of  wine  and  water  in  the  ratio 
9  to  19.  Find  the  ratio  of  the  amounts  taken  from  each 
vessel. 

31.  The  second  of  three  numbers  is  a  mean  proportional 
between  the  other  two.  The  third  number  exceeds  the  sum 
of  the  other  two  by  15,  and  the  sum  of  the  first  and  third 
exceeds  twice  the  second  by  12.     Find  the  numbers. 


VARIATION.  287 


XXIX.  VARIATION. 

323.  One  quantity  is  said  to  vary  directly  as  another 
when  the  ratio  of  any  two  values  of  the  hrst  is  equal  to 
the  ratio  of  the  corresponding  values  of  the  second. 

Note.  It  is  customary  to  omit  the  word  "directly,"  and  say 
simply  that  one  quantity  varies  as  another. 

324.  Let  us  suppose,  for  example,  that  a  workman 
receives  a  fixed  sum  per  day. 

The  amount  which  he  receives  for  m  days  will  be  to 
the  amount  which  he  receives  for  n  days  as  m  is  to  n. 

That  is,  the  ratio  of  any  two  amounts  received  is  equal  to 
the  ratio  of  the  corresponding  numbers  of  days  worked. 

Hence,  the  amount  which  the  workman  receives  varies  as 
the  number  of  days  during  which  he  works. 

325.  One  quantity  is  said  to  vary  inversely  as  another 
when  the  first  varies  directly  as  the  reciprocal  of  the  second. 

Thus,  the  time  in  which  a  railway  train  will  traverse  a 
fixed  route  varies  inversely  as  the  speed;  that  is,  if  the 
speed  be  doubled^  the  train  will  traverse  its  route  in  one- 
half  t\iQ  time. 

326.  One  quantity  is  said  to  vary  as  two  others  jointly 
when  it  varies  directly  as  their  product. 

Thus,  the  wages  of  a  workman  varies  jointly  as  the 
amount  which  he  receives  per  day,  and  the  number  of 
days  during  which  he  works. 

327.  One  quantity  is  said  to  vary  directly  as  a  second 
and  inversely  as  a  third,  when  it  varies  jointly  as  the 
second  and  the  reciprocal  of  the  third. 

Thus,  in  physics,  the  attraction  of  a  body  varies  directly 
as  the  quantity  of  matter,  and  inversely  as  the  square  of 
the  distance. 


288  ALGEBRA. 

328.  The  symbol  oc  is  read  ^'varies  as''-,  thus,  aozb  is 
read  "a  varies  as  b.'" 

329.  If  xccy,  then  x  is  equal  to  y  multiplied  by  a  constant 
quantity. 

Let  x'  and  y'  denote  a  fixed  pair  of  corresponding  values 
of  X  and  y,  and  x  and  y  any  other  pair. 
Then  by  the  definition  of  §  323, 

X      y  xJ 

—  =  —.,  or  x  =  —y. 
x'     y''  y' 

x' 
Denoting  the  constant  ratio  —  by  m,  we  have 

y' 

X  =  my. 

330.  It  follows  from  §§  325,  326,  327,  and  329  that : 

1.  Ifx  varies  inversely  as  y,  x  =  — 

y 

2.  If  X  varies  jointly  as  y  and  z,  x  =  myz. 

3.  If  X  varies  directly  as  y  and  inversely  as  z,  x  = 

331.  Problems  in  variation  are  readily  solved  by  convert- 
ing the  variation  into  an  equation  by  aid  of  §§  329  or  330. 

PROBLEMS. 

332.  1.  If  a;  varies  inversely  as  y,  and  is  equal  to  9  when 
y  =  S,  what  is  the  value  of  x  when  y  =  1S? 

If  X  varies  inversely  as  y,  we  have  x  =  —  (§  330).  p " 

y 


(\M- 


Putting  X  =  9  and  y  =  S,  we  obtain  9  =  — ,  or  m  =  72. 

8 

Then,  x  =  — ;  and  if  y  =  IS,  x=  —  =  4,  Ans.  % 

y  18 

2.  Given  that  the  area  of  a  triangle  varies  jointly  as  its 
base  and  altitude,  what  will  be  the  base  of  a  triangle  whose 
altitude  is  12,  equivalent  to  the  sum  of  two  triangles  whose 
bases  are  10  and  6,  and  altitudes  3  and  9,  respectively  ? 

Let  B,  H,  and  A  denote  the  base,  altitude,  and  area,  respectively, 
of  any  triangle,  and  B'  the  base  of  the  required  triangle. 


Vf 


Oy'A 


V      > 


'A\ 


m 


oX^^,^^ 


VARIATION.  289 

Since  A  varies  jointly  as  B  and  if,  we  have  A  =  mBH  (§  330) . 
Then  tlie  area  of  the  fii-st  triangle  is  m  x  10  x  3,  or  30  m,  and  the 
area  of  the  second  is  ?>i  x  6  x  9,  or  54  m. 

Whence,  the  area  of  the  required  triangle  is  30  m  -+-  54  wi,  or  84  m. 
But  the  area  of  the  required  triangle  is  also  m  x  JS'  x  12. 
Therefore,  12  mB'  =  84  m,  and  B'  =  7,  Ans. 

3.  li  y^  X,  and  is  equal  to  40  when  x  =  5,  what  is  its 
value  when  a;  =  9  ?  ^ 

4.  li  ycc  ^,  and  is  equal  to  48  when  2  =  4,  what  is  the 
expression  for  y  in  terms  of  z^  ? 

5.  If  a;  varies  inversely  as  y^  and  is  equal  to  ^  when 
y  =  ^,  what  is  the  value  of  y  when  a;  =  f  ? 

6.  If  z  varies  jointly  as  x  and  y,  and  is  equal  to  f  when 
y  =  *  and  a;  =  j,  find  the  value  of  z  when  x  =  ^  and  2/  =  f  • 

7.  If  X  varies  directly  as  y  and  inversely  as  z,  and  is 
equal  to  ^  when  y  =  27  and  2;  =  64,  what  is  the  value  of 
X  when  y  =  9  and  2  =  32  ? 

8.  If  5  a;  H-  8  oc  6  2/  —  1,  and  a;  =  6  when  ?/  =  —  3,  what 
is  the  value  of  x  when  y=7? 

9.  If  a;*  Qc  y^,  and  a;  =  4  when  ^  =  4,  what  is  the  value 
of  y  when  x  =  ^? 

10.  The  distance  fallen  by  a  body  from  a  iDOsition  of  rest 
varies  as  the  square  of  the  time  during  which  it  falls.  If  it 
falls  257 J  feet  in  4  seconds,  how  far  will  it  fall  in  6  seconds  ? 

11.  Two  quantities  vary  directly  and  inversely  as  x, 
respectively.  If  their  sum  equals  —  |i  when  a;  =  1,  and 
—  ^  when  a;  =  —  2,  what  are  the  quantities  ? 

12.  The  area  of  a  circle  varies  as  the  square  of  its  diame- 
ter. If  the  area  of  a  circle  whose  diameter  is  4  is  ^^-j  what 
will  be  the  diameter  of  a  circle  whose  area  is  y^  ? 

13.  If  the  volume  of  a  pyramid  varies  jointly  as  its  base 
and  altitude,  find  the  base  of  a  pyramid  whose  altitude  is 
11,  equivalent  to  the  sum  of  two  pyramids,  whose  bases  are 
13  and  14,  and  altitudes  6  and  7,  respectively. 


290  ALGEBRA. 

14.  Given  that  y  is  equal  to  the  sum  of  two  quantities 
which  vary  directly  as  x^  and  inversely  as  x,  respectively. 
If  y  =  —  ^  when  x  =  l,  and  y  =  -^J-  when  x  =  —  2,  what  is 
the  value  of  y  when  a;  =  —  i  ? 

15.  Three  spheres  of  lead  whose  radii  are  6,  8,  and  10 
inches,  respectively,  are  melted  and  formed  into  a  single 
sphere.  Find  its  radius,  having  given  that  the  volume  of 
a  sphere  varies  as  the  cube  of  its  radius. 

16.  The  volume  of  a  cone  of  revolution  varies  jointly 
as  its  altitude  and  the  square  of  the  radius  of  its  base. 
If  the  volume  of  a  cone  whose  altitude  is  3  and  radius  of 
base  5  is  ^^,  what  will  be  the  radius  of  the  base  of  a 
cone  whose  volume  is  ^f ^  and  altitude  5  ? 


17.  If  7  men  in  4  weeks  can  earn  $  238,  how  many  men 
will  earn  $  127^  in  3  weeks ;  it  being  given  that  the  amount 
earned  varies  jointly  as  the  number  of  men,  and  the  number 
of  weeks  during  which  they  work  ? 

18.  If  the  volume  of  a  cylinder  of  revolution  varies 
jointly  as  its  altitude  and  the  square  of  its  radius,  what 
will  be  the  radius  of  a  cylinder  whose  altitude  is  3,  equiva- 
lent to  the  sum  of  two  cylinders  whose  altitudes  are  5  and 
7,  and  radii  6  and  3,  respectively  ? 

19.  If  the  illumination  from  a  source  of  light  varies  in- 
versely as  the  square  of  the  distance,  how  much  farther 
from  a  candle  must  a  book,  which  is  now  15  inches  off,  be 
removed,  so  as  to  receive  just  one- third  as  much  light  ? 

20.  Given  that  y  is  equal  to  the  sum  of  three  quantities, 
the  first  of  which  is  constant,  and  the  second  and  third  vary 
as  X  and  a^,  respectively.  If  y  =  —  19  when  x  =  2,  2/  =  4 
when  a;  =  1,  and  y  =2  when  a;  =  —  1,  what  is  the  expres- 
sion for  y  in  terms  of  a;? 

(Represent  the  constant  by  I,  and  the  other  two  quantities  by  mx 
and  nac^.) 


PROGRESSIONS.  291 

XXX.    PROGRESSIONS. 

ARITHMETIC  PROGRESSION. 

333.  An  Arithmetic  Progression  is  a  series  of  terms  each 
of  which  is  derived  from  the  preceding  by  adding  a  con- 
stant quantity  called  the  common  difference. 

Thus,  1,  3,  5,  1,  9,  11,  •••  is  an  arithmetic  progression  in 
which  the  common  difference  is  2. 

Again,  12,  9,  6,  3,  0,  —3,  •••  is  an  arithmetic  j)rogression 
in  which  the  common  difference  is  —  3. 

334.  Given  the  first  term,  a,  the  common  difference,  d,  and 
the  number  of  terms,  n,  to  find  the  last  term,  I. 

The  progression  is  a,  a  -\-d,  a  -f  2  d,  a  4-  3  c?,  •  •  • . 
It  will  be  observed  that  the  coelficient  of  d  in  any  term 
is  1  less  than  the  number  of  the  term. 

Then  in  the  ?ith  or  last  term  the  coefficient  of  d  is  n  —  1. 
That  is,  l  =  a-\-(n-  1)  d.  (I.) 

335.  Given  the  first  term,  a,  the  last  term,  I,  and  the  num- 
ber of  terms,  n,  to  find  the  smn  of  the  terms,  S. 

JS  =  a  +(a  +  d)-\-(a  +  2  d)+  ••.  +(l-d)-\-l      I 
Writing  the  terms  in  reverse  order, 

S  =  l-\-(l-d)-\-{l-2d)-\-  ...  -}-(a4-fZ)+a.       ^ 
Adding  these  equations  term  by  term, 

2>Sf=(a-f-0  +  (a  +  0  +  («  +  0+  •••  +(a  +  0-h(«  +  0- 
Therefore,     2S  =  n(a  +  I),  and  S  =  '^ (a\  f)-  (H-) 


336.   Substituting  in  (II.)  the  value  of  I  from  (I.),  we 
have  /S  =  ^[2  a  -f  (n  -  l)c^]. 


292  ALGEBRA. 


EXAMPLES. 

337.  1.  Find  the  last  term  and  the  sum  of  the  terms  of 
the  progression  8,  5,  2,  •••  to  27  terms. 

In  this  case,  a  =  8,  d  =  5  —  8  =  —  3,  and  n  =  27. 
Substituting  in  (I.),   Z  =  8  +  (27  -  1)(- 3)=  8  -  78  =  -  70. 

Substituting  in  (II.),  ;S'  =  —(8  _  70)  =  27  x  (  -  31)  =  -  837. 
Z 

Note.  The  common  difference  may  be  found  by  subtracting  the 
first  term  from  the  second,  or  any  term  from  the  next  following  term. 

Find  the  last  term  and  the  sum  of  the  terms  of : 

2.  3,  9,  15,  •••  to  12  terms. 

3.  -7,   -12,   -17,  •••  to  15  terms. 

4.  -69,  -62,  -55,  •.•  to  16  terms. 

5.  I   -|  -|  -  to  17  terms. 

6.  I  |,  g,  ...to  13  terms. 

7.  -^,h^y"'  to  22  terms. 

3'   2'  3' 

3        5        11 

8.  --,  --,  -— ,  ...  to  55  terms. 

9.  -%  -%  -%,  ...  to  19  terms. 

5        2        5 

10.  2a-56,  6a-26,  lOa  +  h,  ...  to  9. terms. 

11.  -^,  I   -^2~'  •"  to  10  terms. 

338.  If  any  three  of  the  five  elements  of  an  arithmetic 
progression  are  given,  the  other  two  may  be  found  by  sub- 
stituting the  given  values  in  the  fundamental  formulae  (I.) 
and  (II.),  and  solving  the  resulting  equations. 


PROGRESSIONS.  293 

5  5 

1.  Given  a  =  —  -,  n  =  20,  S  =  —  -;  find  d  and  /. 

o  o 

Substituting  the  given  values  in  (II.)  >  we  have 

_§=10f_^  +Z  V-or  -^  =  -^+Z;  whence,  Z  =  ^  - ^  =  ?• 
3  V     3         /  6         3        '  3     6     2 

Substituting  the  values  of  Z,  a,  and  w  in  (I.),  we  have 

§.  =  -^^igd;  whence,  19^=-  +  ^  =  ^,  and  d  =  h 
2         3  2     3      6  6 

2.  Given  d  =  -  3,  /  =  -  39,  ^S  =  -  264;  find  a  and  n. 
Substituting  in  (L),  -  39  =  a  +  (w  -  1)(-  3),  or  a  =  37i  -  42.  (1) 
Substituting  the  values  of  S,  a,  and  /  in  (II.),  we  have 

-264=-(3n-42-39),  or  -528=3  n2-81  n,  or  w2-27  n=-176. 

wi.^«o^  27  ±  \/729  -  704      27  ±  6      i^^,  n 

Whence,        n  =  — == =  — == —  =  it)  or  il. 

2  2 

Substituting  in  (1),  a  =  48  -  42  or  33  -  42  =  6  or  -  9. 
Therefore,  a  =  6  and  «  =  16  ;  or,  a  =  —  9  and  71  =  11,  ^ns. 

Note  1.     The  interpretation  of  the  two  answers  is  as  follows  : 
If  a  =  6  and  n  =  16,  the  progression  is 

6,  3,  0,  -  3,  -  6,  -  9,  -  12,  -  15,  -  18,  -  21,  -  24,   -  27,   -  30, 
_  33,  _  36,  -  39. 

If  a  =  —  9  and  n  =  11,  the  progression  is 
-  9,  -  12,  -  15,  -  18,  -  21,  -  24,  -  27,  -  30,  -  33,  -  36,  -  39. 
In  each  of  these  the  sum  is  —  264. 

113 

3.  Given  a  =  -,  ^  —  —  zr^^  ^  —  ~7y''>  ^^^  ^  ^^^  ^' 


Substituting  in  (I.),  ^  =  |  +  (^^  "  1)  (  "  ^  (^^*  ^^^ 

Substituting  the  values  of  S,  «,  and  I  in  (II.),  we  have 

_3^n/l      5-_n\     ^^  _  3  ^  J9-^A     orn-^-9n  =  36. 
2      2V3         12    r        -  V    12    ] 

Whence,  n  ^  9  ±  VsTTTii  ^  9_^  ^  ,2  or  -  3. 


294  ALGEBRA. 

The  value  w=  — 3  is  inapplicable,  for  the  number  of  terms  in  a 

progression  must  be  a  positive  integer. 

Substituting  the  value  7i  =  12  in  (1),  1=  ^^^li^  =  _  -1 . 

7 

Therefore,  I  = and  n  =  12,  Ans. 

12 

Note  2.     A  negative  or  fractional  value  of  n  is  inapplicable,  and 
must  be  rejected,  together  with  all  other  values  dependent  upon  it. 


EXAMPLES. 

4.  Given  d  =  5,  1  =  71,  7i  =  15 ;  find  a  and  S. 

5.  Given  d  =  -  4,  n  =  20,  aS  =  -  620 ;  find  a  and  /. 
-^    6.    Given  a  =  -  9,  n  =  23,  1  =  57;  find  d  and  S. 

7.  Given  a  =  —  5,  n  =  19,  S  =  —  950 ;  find  d  and  I. 

8.  Given  a  =  -,  1  =  —,  8  =  ^^;  find  d  and  n. 

4  4  2 

3 

--   9.    Given  1  =  — ,  ?i  =  19,  8  =  0;  find  a  and  d 
5 

10.  Given  d  =  —,  S  =  —,  a  =  -;  find  I  and  w. 

1^  o  o 

15  1 

11.  Given  a  =  -,  1  =  ——,  d  =  —  —  ;  find  n  and  >S'. 

^  -LJ.  ^^ 

—  12.   Given  c2  =  i,  n  =  17,  S=17;  find  a  and  /, 

z 

13.  Given  Z  =  6,  d  =  -,  aS  =  24;  find  a  and  n. 

6 

14.  Given  Z  =  -5i   n  =  21,  ^  =  -  38i ;  find  a  and  d 
■"-^  16.   Given  a  = ,  /  =  — ^,  S  =  —91 ;  find  d  and  ii. 

16.  Given  a  =  -,  n  =  15,  8  =  ^7— ;  find  d  and  I. 

4  8 

17.  Given  a  =  —-,  d  = ,  >S'  =  -^ ;  find  w  and  I. 


PROGRESSIONS.  295 

18.    Given^  =  -|  d==-^,  S  =  -^',  find  «  and  n. 

-^    19.    Given  «=  5,  c^  =  -|,  /S'  =  -80;  find  ?i  and  Z. 

o 

From  (I.)  and  (II.),  general  formulce  for  the  solution  of 
examples  like  the  above  may  be  readily  derived. 

20.   Given  a,  d,  and  S ;  derive  the  formula  for  n. 

By  §  336,  2  ^  =  w[2 a  +  (n  -  1)  d],  or  dn^  +(2a-d)n  =  2S. 
This  is  a  quadratic  in  ?i ;  and  may  be  solved  by  the  method  of  §  261. 
Multiplying  by  4(Z,  and  adding  (2  a  -  d)2  to  both  members, 
4  dhi^  +  4  d(2  a  -  d)n  +  (2  a  -  rf)2  =  8  d^  +  (2  a  -  d)\ 
Extracting  the  square  root, 


2dw  +  2a  -  d  =  ±y/%dS  +  {2a  -  dy. 


Whence,       n  =  d -2a  ±VSdS -H^a  -  d)^  ^„^ 
2d 

21.  Given  a,  I,  and  n ;  derive  the  formula  for  d. 

22.  Given  a,  n,  and  S]  derive  the  formulae  for  d  and  /. 

23.  Given  c?,  ?i,  and  iS' ;  derive  the  formulae  for  a  and  I. 

24.  Given  a,  d,  and  Z ;  derive  the  formulae  for  n  and  S.  ^ 

25.  Given  cZ,  i,  and  »i ;  derive  the  formulae  for  a  and  S. 

26.  Given  /,  n,  and  ^S' ;  derive  the  formulae  for  a  and  d. 

27.  Given  a,  d,  and  S ;  derive  the  formula  for  I. 

28.  Given  a,  I,  and  aS  ;  derive  the  formulae  for  d  and  n. 

29.  Given  c?,  ?,  and  S ;  derive  the  formulae  for  a  and  ?i. 

339.    To  insert  any  immher  of  arithmetic  means  between 
two  given  terms. 

1.   Insert  5  arithmetic  means  between  3  and  —  5. 

We  are  to  find  an  arithmetic  progression  of  7  terms,  whose  first 
term  is  3,  and  last  term  —  5. 


296  ALGEBRA. 

Tutting  a  =  S,  I  =  -  6,  and  n  =  7,  in  (I.),  §  334,  we  have 

4 
3" 


4 

—  5  =  3  +  6 (Z ;    whence,  6cl  =—  S,  and  d  = 


Hence,  the  required  progression  is 


3,  -,   -,   —1,   — -,   — — ,   —5,  Ans. 
3    3  3         3 


EXAMPLES. 

2.  Insert  6  arithmetic  means  between  3  and  8. 

10  5 

3.  Insert  4  arithmetic  means  between  —  and  —  -• 


4.  Insert  5  arithmetic  means  between and  1. 

o 

3  9 

5.  Insert  7  arithmetic  means  between and  — 

2  2 

6.  Insert  8  arithmetic  means  between  — -  and  —  5. 


3 

7.   Insert  9  arithmetic  means  between  -  and  —  11. 

340.   Let  X  denote  the  arithmetic  mean  between  a  and  6. 
Then,  by  the  nature  of  the  progression, 

X  —  a  —  h  —  X,  or  2x  =  a-\-h. 

Whence,  ^  =  ^^' 

That  is,  tlie  antlimetic  mean  between  two  quantities  is  equal 
to  one-half  their  sum. 

EXAMPLES. 

.  Find  the  arithmetic  mean  between : 

i5.,3  o2a  — 1-,  2a  +  l 

1.  —  and  —■^-  3. and  - — -^-• 

12  20  2a  +  l  2a-l 

2.  (x  +  ly  and  (x  -  7)1  4.    ^^-±-^  and  -  4^!' 

^  a  —  6  a^  —  If 


PROGRESSIONS.  297 


PROBLEMS. 

341.   1.  The  sixth  term  of  an  arithmetic  progression  is 

5  16 

-,  and  the  fifteenth  term  is  —     Find  the  first  term. 

6  3  ^  ^ 

i^  By  §  334,  the  sixth  term  is  a-\-  &d,  and  the  fifteenth  term  a  +  14  d. 

\a+    5d=|  (1) 

Then  by  the  conditions,    I  _ 

\a+Ud  =  ^±  (2) 

I  o 

Q  1 

Subtracting  (1)  from  (2),  9d  =  -  ;  whence,  d  =  — 

2  ^ 

Substituting  in  (1),  a  +  -  =  -  ;  whence,  a  =  —  -,  Ans. 

2     6  3 

2.  Find  four  numbers  in  arithmetic  progression  such  that 
the  product  of  the  first  and  fourth  shall  be  45,  and  the 
product  of  the  second  and  third  77. 

Let  the  numbers  be  x  —  3 y,  x  —  y,  x  +  y,  and  x  +  Sy. 

/•  x^  —  9  v'-^  =  45 

Then  by  the  conditions,  <    ^  2  —  77 

K  X         y  —  II, 

Solving  these  equations,  x=9,  y=±2  ;  or,  a;=  —  9,  y  =±2  (§  276). 

Then  the  numbers  are  3,  7,  11,  15  ;  or,  —  3,  —  7,  —  11,  —  15. 

Note.  In  problems  like  the  above,  it  is  convenient  to  represent 
the  unknown  quantities  by  symmetrical  expressions. 

Thus,  if  five  numbers  had  been  required  to  be  found,  we  should 
have  represented  them  by  x  —  2  y,  x  —  y,  x,  x-\-  y,  and  z  -\-2y. 

3.  Find  the  sum  of  all  the  integers  beginning  with  1 
and  ending  with  100. 

4.  Find  the  sum  of  all  the  even  integers  beginning  with 
2  and  ending  with  1000. 

5.  The  8th  term  of  an  arithmetic  progression  is  10,  and 
the  14th  term  is  —  14.     Find  the  23d  term. 

6.  Find  four  numbers  in  arithmetic  progression  such 
that  the  sum  of  the  first  two  shall  be  12,  and  the  sum  of 
the  last  two  -  20. 


298  ALGEBRA. 

7.  Find  the  sum  of  the  first  15  positive  integers  which 
are  multiples  of  7. 

8.  The  19th  term  of  an  arithmetic  progression  is  9  a;— 2  y, 
and  the  31st  term  is  ISx  —Sy.  Find  the  sum  of  the  first 
thirteen  terms. 

9.  Find  four  integers  in  arithmetic  progression  such  that 
their  sum  shall  be  24,  and  their  product  945. 

10.  How  many  positive  integers  of  three  digits  are  there 
which  are  multiples  of  9  ? 

11.  Find  the  sum  of  all  positive  integers  of  three  digits 
which  are  multiples  of  11. 

12.  The  7th  term  of  an  arithmetic  progression  is  —  I,  the 
16th  term  is  ^,  and  the  last  term  is  ^-.  Find  the  number 
of  terms. 

13.  The  sum  of  the  2d  and  6th  terms  of  an  arithmetic 
progression  is  —  f ,  and  the  sum  of  the  5th  and  9th  terms  is 
—  10.     Find  the  first  term. 

14.  Find  five  numbers  in  arithmetic  progression  such 
that  the  sum  of  the  second,  third,  and  fifth  shall  be  10,  and 
the  product  of  the  first  and  fourth  —  36. 

15.  If  m  arithmetic  means  be  inserted  between  a  and  b, 
what  is  the  first  mean  ? 

16.  How  many  positive  integers  of  one,  two,  or  three 
digits  are  there  which  are  multiples  of  8  ? 

17.  How  many  arithmetic  means  are  inserted  between  4 
and  36,  when  the  second  mean  is  to  the  first  as  4  is  to  3  ? 

18.  A  man  travels  3  miles  the  first  day,  6  miles  the 
second  day,  9  miles  the  third  day,  and  so  on.  After  he  has 
travelled  a  certain  number  of  days,  he  finds  his  average 
daily  distance  to  be  46^  miles.  How  many  days  has  he  been 
travelling  ? 


PROGRESSIONS.  299 

19.  How  many  arithmetic  means  are  inserted  between  | 
and  —  ^,  when  the  sum  of  the  first  two  is  -^  ? 

20.  After  A  had  travelled  for  4i  hours  at  the  rate  of  5 
miles  an  hour,  B  set  out  to  overtake  him,  and  travelled  3 
miles  the  first  hour,  3^  miles  the  second  hour,  4  miles  the 
third  hour,  and  so  on;  in  how  many  hours  will  B  over- 
take A  ? 

21.  Find  three  numbers  in  arithmetic  progression  such 
that  the  sum  of  their  squares  is  347,  and  one-half  the  third 
number  exceeds  the  sum  of  the  first  and  second  by  4i 

22.  The  digits  of  a  number  of  three  figures  are  in  arith- 
metic progression ;  the  sum  of  the  first  two  digits  exceeds 
the  third  by  3;  and  if  396  be  added  to  the  number,  the 
digits  will  be  inverted.     Find  the  number. 

GEOMETRIC   PROGRESSION. 

342.  A  Geometric  Progression  is  a  series  of  terms  each 
of  which  is  derived  from  the  preceding  by  multiplying  by  a 
constant  quantity  called  the  ratio. 

Thus,  2,  6,  18,  54,  162,  •••  is  a  geometric  progression  in 
which  the  ratio  is  3. 

Again,  9,  3,  1,  ^,  ^,  •••  is  a  geometric  progression  in  which 
the  ratio  is  J. 

Negative  values  of  the  ratio  are  also  admissible. 

Thus,  —  3,  6,  —  12,  24,  —  48,  •••  is  a  geometric  progression 
in  which  the  ratio  is  —  2. 

343.  Given  the  first  term,  a,  the  ratio,  r,  and  the  number 
of  terms,  n,  to  find  the  last  term,  I. 

The  progression  is  a,  ar,  ai^,  ai^,  •••. 
It  will  be  observed  that  the  exponent  of  r  in  any  term  is 
1  less  than  the  number  of  the  term. 

Then  in  the  nih.  or  last  term  the  exponent  of  r  is  w  —  1. 
That  is,  I  =  ar^^-K  (I.) 


300  ALGEBRA. 

344.    Given  the  first  term,  a,  the  last  term,  I,  and  the  ratio, 
r,  to  find  the  sum  of  the  terms,  S. 

>S'  =  a  -f  ar  +  ar'^  H h  f«*"~^  +  ct?*""^  +  a>'""^ 

Multiplying  each  term  by  r,  we  have 

rS  =  ar  +  ar^  +  ar^-\ \-  af"  +  a>'"r^  +  ar\ 

Subtracting  the  first  equation  from  the  second, 

rjSI—S  =  ar""  —  a.-^y 

Whence,  ^  ^ar^  -  a 

r  —  1 
But  by  (I.),  §  343,  rl  =  ar\ 

Therefore,  ;S  =  ^^^-=-^.  (II.) 


EXAMPLES.  \ 

345.   1.  Find  the  last  term  and  the  sum  of  the  terms  of 
the  progression  3,  1,  -,  •••  to  7  terms. 

o 

In  this  case,  a  =  3,  r  =  -,  and  n  =  l. 

3 


Substituting  in  (I.),      l  =  ^{^\'  =  l  =  A 


243 

ixJ--3     -L- 3      -2186 
o  V  .-.  *•       •     /TT  N      cr     3      243            729                  729       1093 
Substituting  in  (II. ),   8  =  — j = —  = 2~  "^  "243  ' 

3"^  "3  -3 

Note.    The  ratio  may  be  found  by  dividing  the  second  term  by  the 
first,  or  any  term  by  the  next  preceding  term. 

2.   Find  the  last  term  and  the  sum  of  the  terms  of  the 

progression  —  2,  6,  —  18,  •••  to  8  terms. 

ft 
In  this  case,   a  =  -  2,  r  = =  —  3,  and  w  =  8. 

—  Zi 

Then,  ^  =  - 2(- 3)7  =  - 2  x  (- 2187)=  4374. 

Ind,  ^^  -  3  X  4374  -(-  2)  ^  -  13122  +  2  ^  3^3^^ 
_3_1  -.4 


PROGRESSIONS.  301 

Find  the  last  term  and  the  sum  of  the  terms  of ; 

3.  1,  3,  9,...  to  8  terms. 

4.  6,  4,  -,  •••  to7  terms. 

5.  —  2,  10,  —  50,  •••  to  o  terms. 

6.  2,  4,  8,..-  to  11  terms. 

7.  -S,  -,--,...  to  9  terms. 

8.  --,  -5,  -10,...  to  10  terms. 

r-  2 

A 

9.  —  5,  2,  — , ...  to  6  terms. 

5 

10.  --,-,--,'••  to  7  terms. 

3'  2'       4' 

11.  ?,!,?...  to  5  terms. 
3'  2'  8' 

12.  _  ?,  3,   -  12, ...  to  6  terms. 

4 

346.  If  any  three  of  the  live  elements  of  a  geometric 
progression  are  given,  the  other  two  may  be  found  by  sub- 
stituting the  given  values  in  the  fundamental  formulae  (I.) 
and  (II.),  and  solving  the  resulting  equations. 

But  in  certain  cases  the  operation  involves  the  solution 
of  an  equation  of  a  degree  higher  than  the  second;  and  in 
others  the  unknown  quantity  appears  as  an  exponent,  the 
solution  of  which  form  of  equation  can  usually  only  be 
affected  by  the  aid  of  logarithms  (§  419). 

In  all  such  cases  in  the  present  chapter,  the  equations 
may  be  solved  by  inspection. 

1.    Given  a  =  —  2,  n  =  5,  Z  =  —  32 ;  find  r  and  S. 
Substituting  the  given  values  in  (I.),  we  have 

—  32  =  -  2  »•* ;  whence,  r*  =  16,  and  r  =  ±  2. 


302  ALGEBRA. 

Substituting  in  (II. )» 

If    r  =  2,   S  =  ^^~^P~^~^^  =-6i+2=~62, 

2i  —  1 

_2- 1  -3 

Therefore,  r  =  2  and  .S  =  -  62  ;  or,  r  =  -  2  and  aS^  =  -  22,  Ans. 

Note  1.     The  interpretation  of  the  two  answers  is  as  follows  : 
If  r  =  2,  the  progression  is  —  2,  —  4,  —  8,  —  16,  —  32,  whose  sum 

is  -  62. 

If  r  =  —  2,  the  progression  is  —  2,  4,  —  8,  16,  —  32,  whose  sum  is 

-22. 

2.  Given  a  =  3,  r  = ,  &  =  —-—  ;  find  n  and  I. 

'  3  729  ' 

-lz-3 

c  ^,  ^-^  *•       •     /TT  N      1640         3  Z  +  9 

Substituting  m  (II. ) ,     -j^  = =  -^. 

~3~ 

Whence,        Z  4-  9  = ;  or,  Z  = 9  = • 

729  729  729 

Substituting  the  values  of  Z,  a,  and  r  in  (I.),  we  have 

'      729        V     3^       '         V     3;  2187 

Whence,  by  inspection,  n  —  1  =  7,  or  w  =  8. 

EXAMPLES. 

3.  Given  r  =  2,  n  =  9,  ^  =  256 ;  find  a  and  >S. 

4.  Given  ?'  =  -,  ?i  =  5,  /S' = ;  find  a  and  I. 

3  '  27  ' 

5.  Given  a  =  -  2,  n  =  6,  Z  =  2048  ;  find  r  and  /S'. 

^    6.    Given  a  =  2,  ?-  = ,  Z  = -:  find  n  and  aS'. 

'  2/  256' 

7.  Given  r  =  i,  71  =  11,  ^  =  ^;  find  a  and  I 

8.  Given  a  =  |,  n  =  9,  ^  =  ^;  find  r  and  /S. 

3  128 


PROGRESSIONS.  303 

9.    Given  a  =  —  S,  1  =  —  —,  S  =  — '——;   find  r  and  ii. 

10.  Given  «==-,  ^*  =  ~o'  '^~i7^'  ^^^  ^  ^^^  ^^• 

11.  Given  /  =  192,  r  =  -2,  .S  =  129;  find  «  and  n. 

12.  Given  «  =  -^j  ^"""199'  "^"""19^'  ^^^  ''  ^"^  '^' 

From  (I.)  and  (II.),  general  formulae  may  be  derived  for 
the  solution  of  cases  like  the  above. 

13.  Given  a,  ?•,  and  S ;  derive  the  formula  for  I. 

14.  Given  a,  Z,  and  S ;  derive  the  formula  for  r. 

15.  Given  ?*,  Z,  and  *S;  derive  the  formula  for  a. 

16.  Given  r,  ?i,  and  Z ;  derive  the  formulie  for  ci  and  S. 

17.  Given  r,  ri,  and  ^5;  derive  the  formulae  for  a  and  /. 

18.  Given  a,  ?i,  and  I ;  derive  the  formulae  for  r  and  aS. 

Note  2.  If  the  given  elements  are  u,  Z,  and  /V,  equations  for  a 
and  r  may  be  found,  but  there  are  no  definite  formuloe  for  their 
values.  The  same  is  the  case  when  the  given  elements  are  a,  w, 
and  8. 

The  general  formulae  for  n  involve  logarithms  ;  these  cases  are 
discussed  in  §  419. 

347.  The  limit  (§  292)  to  which  the  sum  of  the  terms  of 
a  decreasing  geometric  progression  approaches,  when  the 
number  of  terms  is  indefinitely  increased,  is  called  the  sum 
of  the  series  to  infinity. 

Formula  (II.),  §  344,  may  be  written 
cr      a  —  rl 

It  is  evident  that,  by  sufficiently  continuing  a  decreasing 
geometric  progression,  the  last  term  may  be  made  numeri- 
cally less  than  any  assigned  number,  however  small. 

Hence,  when  the  number  of  terms  is  indefinitely  increased, 
I,  and  therefore  rl,  approaches  the  limit  0. 


304  ALGEBRA. 

Then  the  fraction  ^  ~  ^    approaches  the  limit  — — 
1  —  r  1  —  r 

Therefore,  the  sum  of  a  decreasing  geometric  progression 
to  infinity  is  given  by  the  formula 

S  =  -^.  (III.) 

1  —  r 

EXAMPLES. 

1.   Find  the  sum  of  the  series  4,  — -,  — ,  •••,  to  infinity. 

o     J 

2 

In  this  case,  a  =  4,  r  = 

4         12 
Substituting  in  (III.),  S  = ^  =  ^,  Ans. 

3 

Find  the  sum  of  the  following  to  infinity : 


2. 

3,  1,  1  .... 

6. 

7   21    63 
4'  32'  256' 

•. 

3. 

16,  -4,1,  .... 

--7. 

2       15 

5'      3'  18' 

.... 

4. 

_1    1    _JL  ... 

'  5'      25'      • 

8. 

1        1 

8'      18' 

2 
81' 

5. 

5       10       20 

3'       9'      27'*"* 

^   9. 

5       5  35 

7'      8'  64' 

.... 

348.    To  find  the  value  of  a  repeating  decimal. 

This  is  a  case  of  finding  the  sum  of  a  decreasing  geometric 
series  to  infinity,  and  may  be  solved  by  formula  (III.). 

1.   Findthe  value  of  .85151.... 

We  have,  .85151 ...  =  .8  -I-  .051  +  .00051  +  .... 

The  terms  after  the  first  constitute  a  decreasing  geometric  pro- 
gression, in  which  a  =  .051  and  ?'  =  .01. 

.051        .051       51        17 


Substituting  in  (III.),  S 


1  -  .01       .99      990     .330 


8        17  281 

Then  the  value  of  the  given  decimal  is 1-  — -,  or  — -,  Ans. 

10      ooO         oov 


PROGRESSIONS.  305 

EXAMPLES. 
Find  the  values  of  the  following : 

2.  .8181....  4.   .69444....  6.   .11567567.... 

3.  .296296....        -5.   .58686....         -7.   .922828-... 

349.  To  insert  any  number  of  geometric  means  between  two 

given  terms. 

128 

1.  Insert  5  geometric  means  between  2  and  — — • 

i  ^u 

We  are  to  fiud  a  geometric  progression  of  7  terms,  whose  first 

128 

term  is  2,  and  last  term  -— • 

'  729 

Putting  a  =  2,  1  =  — ,  and  n  =  7,  in  (I.),  §  343,  we  have 

1^  =  2r6 ;  whence,  r"^  =  — ,  and  r  =  ±  ^. 
729  '  729  3 

Hence,  the  required  result  is 

o     j_4     8     ^16     32      ,    64      128     .^ 
^'    ^3'    9'    ^27'    81'    ^24"3'    729'  '^'''' 

EXAMPLES. 

2.  Insert  4  geometric  means  between  3  and  729. 

1  64 

3.  Insert  6  geometric  means  between  -  and  —  — ■• 

6  o 

4.  Insert  5  geometric  means  between  2  and  128. 

2  125 

5.  Insert  3  geometric  means  between  —  -  and -— 

5  o 

6.  Insert  4  geometric  means  between  —  -  and  3584. 

z 

243  2 

7.  Insert  7  geometric  means  between  — -  and  — • 

350.  Let  X  denote  the  geometric  mean  between  a  and  h. 

Then,  by  the  nature  of  the  progression,  -  =  -,  or  a^  =  ab. 

a     X 


306  ALGEBRA. 

Whence,  x  —  Vab. 

That  is,  the  geometric  mean  between  tivo  quantities  is  equal 
to  the  square  root  of  their  product. 


EXAMPLES. 

Find  the  geometric  mean  between : 

1.   2ijandlif.  2.    9  +  4V5  and  9  -  4 VS. 

3.   a-  +  2  ab  +  b^  and  a^-2  ah  +  b\ 

.     2x^  +  4:xy  xy  -\-2  y- 

xy  —  2y^  2x^  —  4:xy 

PROBLEMS. 

351.   1.  Find  three  numbers  in  geometric  progression  such 
that  their  sum  shall  be  14,.  and  the  sum  of  their  squares  84. 

Let  the  numbers  be  a,  ar,  and  ar^. 


i       a  +  ar  +  ar2  =  14. 
Then  by  the  conditions,   \ 

(1) 
(2) 

Dividing  (2)  by  (1),                a  -  ar -Y  ar'^  =  Q. 

(3) 

Subtracting  (3)  from  (1),                      2  ar  =  8,  or  r  =  -• 

a 

(4) 

Substituting  in  (1),  a  +  4  +  —  =  14,  or  a2  _  10  a  =  -  16. 
a 

Solving  this  equation,               a  =  8  or  2. 

Substituting  in  (4),                   r  =  |  or  |  =  1  or  2.    ' 

8        2      2 

Therefore,  the  numbers  are  2,  4,  and  8,  Ans. 

2.  The  4th  term  of  a  geometric  progression  is  —  ^-,  and 
the  7th  term  is  |^f  f .     Find  the  second  term. 

3.  The  sum  of  the  first  and  last  of  four  numbers  in  geo- 
metric progression  is  112,  and  the  sum  of  the  second  and 
third  is  48.     Find  the  numbers. 

4.  The  product  of  three  numbers  in  geometric  progres- 
sion is  — 1000,  and  the  sum  of  the  squares  of  the  second 
and  third  is  500.     Find  the  numbers. 


PROGRESSIONS.  307 

5.  A  man  saves  every  year  half  as  much  again  as  he 
saved  the  preceding  year.  If  he  saved  $  128  the  first  year, 
to  what  sum  will  his  savings  amount  at  the  end  of  seven 
years  ? 

6.  A  body  moves  12  feet  the  first  second,  and  in  each 
succeeding  second  five-eighths  as  far  as  in  the  preceding 
second,  until  it  comes  to  rest.    How  far  will  it  have  moved  ? 

7.  The  5th  term  of  a  geometric  progression  is  —  f ,  and 
the  9th  term  is  —  ^^.     Find  the  11th  term. 

8.  If  m  geometric  means  be  inserted  between  a  and  b, 
what  is  the  first  mean  ? 

9.  The  sum  of  three  numbers  in  arithmetic  progression 
is  12.  If  the  first  number  be  increased  by  5,  the  second  by 
2,  and  the  third  by  7,  the  resulting  numbers  form  a  geo- 
metric progression.     What  are  the  numbers  ?*" 

10.  Divide  $  700  between  A,  B,  C,  and  D,  so  that  their 
shares  may  be  in  geometric  progression,  and  the  sum  of  A's 
and  B's  shares  equal  to  $  252. 

11.  There  are  four  numbers,  the  first  three  of  which  form 
an  arithmetic  progression,  and  the  last  three  a  geometric 
progression.  The  sum  of  the  first  and  third  is  2,  and  of 
the  second  and  fourth  37.     What  are  the  numbers  ? 

12.  Find  che  ratio  of  the  geometric  progression  in  which 
the  sum  of  the  first  ten  terms  is  244  times  the  sum  of  the 
first  five  terms. 

13.  There  are  three  numbers  in  geometric  progression 
whose  sum  is  19.  If  the  first  be  multiplied  by  |,  the  second 
by  |,  and  the  third  by  -f,  the  resulting  numbers  form  an 
arithmetic  progression.     What  are  the  numbers  ? 

HARMONIC  PROGRESSION. 

352.  A  Harmonic  Progression  is  a  series  of  terms  whose 
reciprocals  form  an  arithmetic  progression. 


308  ALGEBRA. 

Thus,  1,  I,  "l^,  f,  i,  •••  is  a  harmonic  progression,  because 
the  reciprocals  of  the  terms,  1,  3,  5,  7,  9,  •••,  form  an  arith- 
metic progression. 

353.  Any  problem  in  harmonic  progression  which  is  sus- 
ceptible of  solution,  may  be  solved  by  taking  the  reciprocals 
of  the  terms,  and  applying  the  formulae  of  the  arithmetic 
progression.  There  is,  however,  no  general  method  for 
finding  the  su7n  of  the  terms  of  a  harmonic  progression. 

354.  Let  X  denote  the  harmonic  mean  between  a  and  h. 

Then,  -  is  the  arithmetic  mean  between  -  and  r  (§  352). 
X  a         b  ^  ^ 

Whence  by  §  340,   -  =  — rr—  =  ,.    ^  ,  and  x  =  — — j- 
•^  '   X         2  2ab'  a  +  5 


EXAMPLES. 

355.   1.   Find  the  last  term  of  the  progression  2,  |,  ■§-,  ••• 
to  36  terms. 

Taking  the  reciprocals  of  the  terms,  we  have  the  arithmetic  pro- 

13    5 
gression  -i   -»   ->   •••. 

Jj     2i     2i 

In  this  case,  a  =  -,  d=\^  and  n  =  36. 

Substituting  in  (I.),  §  334,  we  have  Z  =  ^-  4-  (36  -  1)  x  1  =  — • 
Taking  the  reciprocal  of  this,  the  last  term  of  the  given  harmonic 
progression  is  — ,  Ans. 

2.   Insert  5  harmonic  means  between  2  and  —  3. 

We  have  to  insert  5  arithmetic  means  between  -  and 

2  3 

Putting  a  =  -,  Z  =  — ,  and  n  —  7,  in  (I.),  §  334,  we  have 
2  3 

_  1  =  1  +  6  cZ ;  whence,  6  d  =  -  -,  or  (Z  =  -  A. 
3      2'  '  6'  36 


PROGRESSIONS.  309 

Then  the  arithmetic  progression  is 


3. 


1     13 

2      1 

1           7 

1 

2'    30' 

y'  12' 

18'        36' 

3' 

Therefore, 

the  required 

harmonic  progressior 

lis 

2    36    9 
'    13'    2' 

12,    - 

18,    -f,    - 

3,  Ans. 

Find  the  last  terms  of  the 

following : 

1    ?4  1 

'   5'  7'  '' 

•  ••  to  13  terms. 

4    4  12 
5' 43' 

12 
71'  *" 

to  25  terms. 

6.    -3,  2,  -,  ...  to  38  terms. 
4 

6.  -|-|-g,  ...to43terms. 

«        5       2       5        ^    ,^  , 

7.  --,__,--,  ...  to  17  terms. 

8.  Insert  6  harmonic  means  between  2  and 

9 

2  2 

9.  Insert  7  harmonic  means  between  —  -  and  — 

5  7 

10.  Insert  8  harmonic  means  between  —  -  and  —  -. 

5  5 

Find  the  harmonic  mean  between : 

11.  3  and  6.  12.  l^H^  and  -  ^  ~  ^ 


1-f-a;  l+ar^ 

13.  The  first  term  of  a  harmonic  progression  is  x,  and  the 
second  term  is  y ;  continue  the  series  to  three  more  terms. 

14.  The  arithmetic  mean  between  two  numbers  is  1,  and 
the  harmonic  mean  —  15.     Find  the  numbers. 

15.  The  5th  term  of  a  harmonic  progression  is  —  f ,  and 
the  11th  term  is  —  ^.     What  is  the  15th  term  ? 

16.  Prove  that,  if  a,  b,  and  c  are  in  harmonic  progression, 

a:  c  =  a  —  b  :b  —  c. 


310  ALGEBRA. 

XXXI.    THE  BINOMIAL  THEOREM. 

POSITIVE  INTEGRAL  EXPONENT. 

356.  The  Binomial  Theorem  is  a  formula  by  means  of 
which  any  power  of  a  binomial,  positive  or  negative,  inte- 
gral or  fractional,  may  be  expanded  into  a  series. 

We  sh3,ll  consider  in  the  "present  chapter  those  cases 
only  in  which  the  exponent  is  a  positive  integer. 

357.  Proof  of  the  Binomial  Theorem  for  a  Positive  Inte- 
gral Exponent. 

By  actual  multiplication,  we  obtain: 

(a -\- xy  =  a^  -\- 2  ax -\-  Qc^  ] 

(a  -{-xy  =  a^-hS  A  -h  3  aa^  +  a^ ;        "* 

(a  -^xy  =  a*-\-4:  a^x  +  6  cv^x^ -^  4  ax^  +  x* ;  etc. 

In  the  above  results,  we  observe  the  following  laws : 

.1.  The  number  of  terms  is  greater  by  1  than  the  expo- 
nent of  the  binomial. 

2.  The  exponent  of  a  in  the  first  term  is  the  same  as  the 
exponent  of  the  binomial,  and  decreases  by  1  in  each  suc- 
ceeding term. 

3.  The  exponent  of  x  in  the  second  term  is  1,  and  in 
creases  by  1  in  each  succeeding  term. 

4.  The  coefficient  of  the  first  term  is  1,  and  the  coefficient 
of  the  second  term  is  the  exponent  of  the  binomial. 

5.  If  the  coefficient  of  any  term  be  multiplied  by  the 
exponent  of  a  in  that  term,  and  the  result  divided  by  the 
exponent  of  x  in  the  term  increased  by  1,  the  quotient  will 
be  the  coefficient  of  the  next  following  term. 


THE   BINOMIAL   THEOREM.  311 

358.  If  the  law«  ui. s  o57  l)e  assuuit-d  to  hold  for  the 
expansion  of  (a  +  x)",  where  n  is  any  positive  integer,  the 
exponent  of  a  in  the  first  term  is  n,  in  the  second  term 
71  —  1,  in  the  third  term  w  —  2,  in  the  fourth  term  n  —  3,  etc. 

The  exponent  of  x  in  the  second  term  is  1,  in  the  third 
term  2,  in  the  fourth  term  3,  etc. 

The  coefficient  of  the  first  term  i&  1 ,'  of  the  second  term  n. 

Multiplying  the  coefficient  of  the  second  term,  n,  by  w— 1, 
the  exponent  of  a  in  that  term,  and  dividing  the  result  by 
the  exponent  of  x  in  the  term  increased  by  1,  or  2,  we  have 
—\- — — ^  as  the  coefficient  of  the  third  term ;  and  so  on. 

Then,  (a  +  a!)»  =  a-.  +  wa'-'as+^i-fe-I^a—V       -f- 

+  !L(!Lzilfc21a-V+....       (1) 
Multiplying  both  members  of  (1)  by  a  +  x,  we  have 

I  J.  •  ^ 

4|a^4^  na-^ar^.  +  ^^  (^^  ~  •^)  aP-^x^  +  •  •  • . 
Collecting  the  terms  which  contain  like  powers  of  a  and  x, 

=  a"+' +  (?.  +  1)  a"a;  +  n  r?i^  +  llo"-'a!' 

*  A  point  i?  often  used  in  place  of  the  sign  x ;  thus,  1  •  2  is  the 
same  as  1  x  2. 


312  ALGEBRA. 


Or,   (a  +  xy+^  =  a"+i  +  {n  -\- 1)  a"x  -f  71  f^Lill  a"-^a^ 


+ 


1.2 


=  a"+i  +  (w  +  1)  a^'x  +  ^^^+^)^  a"  ^  V 
1  •  ^ 

I  (^^  +  l)^(n-l)       2  , 

1.2.3  * 

It  will  be  observed  that  this  result  is  in  accordance  with 
the  laws  of  §  357 ;  which  proves  that,  if  the  laws  of  §  357 
hold  for  any  power  oi  a  +  x  whose  exponent  is  a  positive 
integer,  they  also  hold  for  a  power  whose  exponent  is 
greater  by  1. 

But  the  laws  have  been  shown  to  hold  for  (a  +  xy,  and 
hence  they  also  hold  for  (a  +  xy ;  and  since  they  hold  for 
(a  +  xy,  they  also  hold  for  (a  +  xy ;  and  so  on. 

Therefore,  the  laws  hold  when  the  exponent  is  any  posi- 
tive integer,  and  equation  (1)  is  proved  for  every  positive 
integral  value  of  7i. 

Equation  (1)  is  called  the  Binomial  Theorem. 

Note  1.  The  above  method  of  proof  is  known  as  Mathematical 
Induction. 

Note  2.  In  place  of  the  denominators  1-2,  1-2 -3,  etc.,  it  is 
usual  to  write  [2,  [3,  etc.  The  symbol  \n,  read  '■'■  factorial  w,"  signifies 
the  product  of  the  natural  numbers  from  1  to  w  inclusive. 

359.  Putting  a  =  1  in  equation  (1),  §  358,  we  have 

(1  +  a;)"  =  1  4-  nx  +  -^TS — ^x^  +  — ^ 7^ ^^  +  •••• 

EXAMPLES. 

360.  In  expanding  expressions  by  the  Binomial  Theorem, 
it  is  convenient  to  obtain  the  exponents  and  coefficients  of 
the  terms  by  aid  of  the  laws  of  §  357,  whi^-b  have  been 
proved  to  hold  for  any  positive  integral  exconefit. 


THE   BINOMIAL   THEOREM.  313 

1.  Expand  (a  -|-  xy. 

The  exponent  of  a  in  the  first  term  is  5,  in  the  second  term  4,  in 
the  third  term  3,  in  the  fourth  term  2,  in  the  fifth  term  1. 

The  exponent  of  x  in  the  second  term  is  1,  in  the  third  term  2,  in 
the  fourth  term  3,  in  the  fifth  term  4,  in  the  sixth  term  5. 

The  coefficient  of  the  first  term  is  1  ;  of  the  second  term,  5. 

Multiplying  the  coefficient  of  the  second  term,  5,  by  4,  the  exponent 
of  a  in  that  term,  and  dividing  the  result  by  the  exponent  of  x  in  the 
term  increased  by  1,  or  2,  we  have  10  as  the  coefficient  of  the  third 
term;  and  so  on. 

Then,  (a  +  x)^  =  a&  +  5  a^^  +  10  aV-^  +  10  «%»  +  5  ax*  +  x^  Ans. 

Note  1.  The  coefl&cients  of  terms  equally  distant  from  the  begin- 
ning and  end  of  the  expansion  are  equal.  Thus  the  coefficients  of  the 
latter  half  of  an  expansion  may  be  written  out  from  the  first  half. 

If  the  second  term  of  the  binomial  is  negativey  it  should 
be  enclosed,  sign  and  all,  in  a  parenthesis  before  applying 
the  laws.  In  reducing  jifterwards,  care  must  be  taken  to 
apply  the  principles  of  §  186. 

2.  Expand  (1  —  xf. 

We  have,  (l-a;)6  =  [l  +  (-x)]« 

=  16+6  .  15 .  (_a;)  + 15  .  1* .  (-  x)2+20  .  13  .  (-x)3 

+  15  .  12  .  (-  a-)*  +  6  . 1  .  (-  xY  +  (-  x)6 
=  1  -  6 X  +  15x2  -  20 x3  +  15 X*  -  6 x^  +  x6,  Ans. 

Note  2.  If  the  first  term  of  the  binomial  is  numerical,  it  is  con- 
venient to  write  the  exponents  at  first  without  reduction.  The  result 
should  afterwards  be  reduced  to  its  simplest  form. 

If  either  term  of  the  binomial  has  a  coefficient  or  exponent 
other  than  unity,  it  should  be  enclosed  in  a  parenthesis  be- 
fore applying  the  laws.  5 —  -^ 
^^^     ^                   _                    rTvu  z.  aA>5 

3.  Expand  (3  m'  -  Vn)\  !{_  o  ^ 

(3  9»2  _  ^„.)4  ^  [  (3  ,^2)  4.  (  _  ^*)  ]4 


=  (3  ?n2)4  +  4(3?n2)3(-  J)  +  6(3  j?i2)2(_  n^)2 

+  4(3 m2)(- 71^)3 +  (-n^)4 
=  81 7/18  -  108  mhi^  +  54  m%*  -  12  m^n  +  n\  Ans. 


314  ALGEBRA. 

Expand  the  following : 

24.    f2a^+-i^Y. 
2aV 


4.  (x  +  iy.         J.    15.    (1+2  my. 

5.  (a  +  xf.  16.    (l-x)«. 


6.  (a-o^y.  ,^  17.  (^0^4  +  ^  1)5^  g^^  /^^2__L 

7.  (m-n)^  ^  18.  (a^- 2/.  ^             ^^' 

8-  a+^-y.  .  19.  (3-f.^y.  ^«-  (2--^-v^)- 

9.  (a -by.  ^^  ^1         ^_^6  ^^  fx-'^      ^5/-3^ 


(-!.»)•  -  (¥-«)• 


10.  (a'-\-b'cy. 

11.  (a.-2'«  +  /«)^.    ^,   2^^    (4at-a;^y.        ^8.    (V^^  +  4^< 

12.  (2a-iy.      '  ,  4X5  /  I        3 

14.   (a -3 by.  23.   (m' +  5 a;-^.  y  30.   (2a -36)^ 

31.    (ah-i  +  a-hhl  32.   (s^I^-2^jlJ. 

A  trinomial  may  be  raised  to  any  power  by  the  Binomial 
Theorem  if  two  of  its  terms  be  enclosed  in  a  parenthesis 
and  regarded  as  a  single  term. 

33.   Expand  (x^-2x-  2y. 

(a:2  _  2 X  -  2)4  =  [ (a:2  _  2  x)  +  ( -  2)]^ 

=  (x^-2  xy  +  4(a;2  -  2  x^i-  2)  +  6(a;2  -  2  x)2( -  2)2 

+  4(^2 -2a;)(- 2)3 +  (-2)4 
=  a:8-8«7  4. 24  ic6_32x5+ 16x4 -8(x6-6xH  12^4-8x5) 

+  24  (x*  -  4  x3  +  4  x2)  -  32  (x2  -  2  x)  +  16 
s«x8-8x7+16x6  4- 1^x5-56x4-32x3 
+  64  x2  +  64  X  +  16,  Ans. 

Expand  the  following : 

34.  (i_a;  +  a^)4.  37.  (a^-2x-Sy. 

35.  (x'  +  x  +  2y.  y38.  (l+x-x^y. 

36.  (l--i-3x-xy.  39.  (x'-x  +  2y. 


THE   BINOMIAL   THEOREM.  315 

361.  To  find  the  rtk  or  general  term  in  the  expansion  of 
(a  -h  xy. 

The  following  laws  will  be  found  to  hold  for  any  term 
in  the  expansion  of  (a  -f  xy\  in  equation  (1),  §  358 : 

1.  The  exponent  of  x  is  less  by  1  than  the  number  of 
the  term. 

2.  The  exponent  of  a  is  n  minus  the  exponent  of  x. 

3.  The  last  factor  of  the  numerator  is  greater  by  1  than 
the  exponent  of  a. 

4.  The  last  factor  of  the  denominator  is  the  same  as  the 
exponent  of  x. 

Hence,  in  the  rth  term,  the  exponent  of  x  is  r  —  1. 
The  exponent  of  a  is  ?i  —  (r  —  1),  or  n  —  r  -f  1. 
The  last  factor  of  the  numerator  is  n  —  r  4-  2. 
The  last  factor  of  the  denominator  is  r  —  1. 
Therefore,  the  ?'th  term  of  the  expansion  of  (a  +  .^•)"  is 
n(n  -  l)(n-2)  ...  (n  -  r  +  2)^._^,^_, 


EXAMPLES. 

362.  In  finding  any  term  of  an  expansion,  it  is  convenient 
to  obtain  the  coefficient  and  the  exponents  of  the  terms  by 
aid  of  the  laws  of  §  361. 

1.   FindtheSth  termof  (3a*-6-^)^\ 

We  have,        (3  a*  -  ft-i)"  =  [(3  a*)  +  (-  6-i)]".. 

In  this  case,  n  =  11  and  r  =  8. 

The  exponent  of  (—  6-i)  is  8  —  1,  or  7. 

The  exponent  of  (3  a^)  is  11  —  Y,  or  4. 

The  first  factor  of  the  numerator  is  11,  and  the  last  factor  4  +  1, 
or  5. 

The  last  factor  of  the  denominator  is  7. 

Hence,  the  8th  term  ^  1^  •  tO  •  ^  •  8  •  7  •  6  •  5  .3  oi)4(_  5-1)7 
1.2.3.4.5.6.7   ^      2'  ^  ^ 

=  330  .  (81  a2)(-  6-")  =  -26730a26-7,  Ans. 


316  ALGEBRA. 

Note.  If  the  second  term  of  the  binomial  is  negative,  it  should  be 
enclosed,  sign  and  all,  in  a  parenthesis  before  applying  the  laws. 

If  either  term  of  the  binomial  has  a  coefficient  or  exponent  other 
than  unity,  it  should  be  enclosed  in  a  parenthesis  before  applying  the 
laws. 


Find  the 

2. 

4th  term  of  (a  +  xf. 

3. 

9th  term  of  (m  +  1)". 

4. 

5th  term  of  (a  -  bf. 

5. 

10th  term  of  (1  -  x^. 

6. 

9th  term  of  (m^  -  ny\ 

7. 

7fhtevmoifa-'  +  ~\ 

8.   4th  term  of 


\y    ^J 

9.   lOth  term  of  (a*"  +  a^'f^ 

10.  8th  term  of  (x-^-  -  2  yfy^ 

11.  6th  term  of  (a~^  -f  3ar*)'°. 


/     3  1       \13 

12.  7th  term  of  f  a^ ^]  • 

\         WbJ 

13.  8th  term  of  (a"^  +  2^xy\ 
/  14.   5th  term  of  (m^  +  ^ 

15.   6th  term  of  fx''  -  ^^  • 


16.   Find  the  middle  term  of 


e-0' 


UNDETERMINED   COEFFICIENTS.  317 

XXXII.  UNDETERMINED  COEFFICIENTS. 

CONVERGENCY  AND  DIVERGENCY  OF  SERIES. 

363.  A  Series  is  a  succession  of  terms  so  related  that 
each  may  be  derived  from  one  or  more  of  the  preceding  in 
accordance  with  some  fixed  law. 

A  Finite  Series  is  one  having  a  finite  number  of  terms. 

An  Infinite  Series  is  one  the  number  of  whose  terms  is 
unlimited. 

The  progressions,  in  general,  are  examples  of  finite  series ; 
but  in  §  347  we  considered  infinite  geometrical  series. 

364.  Infinite  series  may  be  developed  by  Division. 
Let  it  be  required,  for  example,  to  divide  1  by  1  —  x. 

l-x)l(l-\-x^x'+  ... 
1-a; 


X 

X  —  x^ 
Therefore,  — i-=  1  +  a;  +  ar^+ ••• . 

1  —X 

Infinite  series  may  also  be  developed  by  Evolution  (see 
Exs.  25  to  30,  §  195),  and  by  other  methods,  one  of  the  most 
important  of  which  will  be  considered  in  §  369. 

365.  A  series  is  said  to  be  convergent  either  when  the 
sum  of  the  first  n  terms  approaches  a  certain  fixed  quantity 
as  a  limit  (§  292)  ;  when  n  is  indefinitely  increased  ;  or  when 
the  sum  of  all  the  terms  is  equal  to  a  fixed  finite  quantity. 

A  series  is  said  to  be  divergent  when  the  sum  of  the  first 
n  terms  can  be  made  to  numerically  exceed  any  assigned 
quantity,  however  great,  by  taking  n  sufficiently  great. 


318  ALGEBRA. 

366.   Consider,  for  example,  the  infinite  series 

I.    Suppose  X  =  Xi,  where  Xi  is  numerically  <  1. 
The  sum  of  the  first  n  terms  is  now 

l  +  x,-{-  X,'  +  •••  +  xr'  =  \^^^  (§  86). 

If  n  is  indefinitely  increased,  Xi"  decreases  indefinitely  in 
absolute  value,  and  approaches  the  limit  0. 

Then  the  fraction approaches  the  limit 


1  —  Xi  1  —  a'l 

That  is,  the  sum  of  the  first  n  terms  approaches  a  certain 
fixed  quantity  as  a  limit,  when  n  is  indefinitely  increased. 
Hence,  the  series  is  convergent  Avhen  x  is  numerically  <  1. 

II.  Suppose  x  =  l. 

In  this  case,  each  term  of  the  series  is  equal  to  1,  and  the 
sum  of  the  first  n  terms  is  equal  to  n ;  and  this  sum  can  be 
made  to  exceed  any  assigned  quantity,  however  great,  by 
taking  n  sufficiently  great. 

Hence,  the  series  is  divergent  when  x  =  l. 

III.  Suppose  x  =  —  l. 

In  this  case,  the  series  takes  the  form  1  —  1  +  1— iH , 

and  the  sum  of  the  first  ii  terms  is  either  1  or  0  according 
as  n  is  odd  or  even. 

Hence,  the  series  is  neither  convergent  nor  divergent  when 
a;  =  -l. 

IV.  Suppose  X  =  iCi,  where  x^  is  numerically  >  1. 
The  sum  of  the  first  n  terms  is  now 

1  +  a^i  +  o^i'  +  -  +  xr'  =  ^^^  (§  86). 

iCj  —  1 

By  taking  n  sufficiently  great,  — —  can  be  made  to 

numerically  exceed  any  assigned  quantity,  however  great. 


UNDETERMINED  COEFFICIENTS.  319 

Hence,  the  series  is  divergent  when  x  is  numerically  >  1. 

367.  Consider  the  infinite  series 

developed  by  the  fraction (§  364), 

i  —  X 

Let  x=  .1,  in  which  case  the  series  is  convergent  (§  366). 

The  series  now  takes  the  form  1  +  -1  +  -01  H-.OOl  +  •••, 
while  the  value  of  the  fraction  is  — ,  or  — • 

In  this  case,  however  great  the  number  of  terms  taken, 
their  sum  will  never  exactly  equal  — ;  but  it  approaches 
this  value  as  a  limit.     (See  §  347.) 

Thus,  if  an  infinite  series  is  convergent,  the  greater  the 
number  of  terms  taken,  the  more  nearly  does  their  sum 
approach  to  the  value  of  the  expression  from  which  the 
series  was  developed. 

Again,  let  x  =  10,  in  which  case  the  series  is  divergent. 

The  series  now  takes  the  form  1  +  10  -f-  100  +  1000  +  .-., 

while  the  value  of  the  fraction  is — ,  or  —  - 

J.  —  J.U  J 

In  this  case  it  is  evident  that,  the  greater  the  number  of 
terms  taken,  the  more  does  their  sum   diverge  from  the 

value  — -• 
9 

Thus,  if  an  infinite  series  is  divergent,  the  greater  the 
number  of  terms  taken,  the  more  does  their  sum  diverge 
from  the  value  of  the  expression  from  which  the  series  was 
developed. 

It  follows  from  the  above  that  an  infinite  series  cannot  be 
used  for  the  purposes  of  demonstration,  unless  it  is  convergent. 

368.  The  infinite  series 

a -\- hx -\- ca?  -\-  dx^  -[-••• 

is  convergent  when  x  =  0',  for  the  sum  of  all  the  terms  is 
equal  to  a  when  x  =  0. 


320  ALGEBRA. 

THE  THEOREM  OF  UNDETERMINED  COEFFICIENTS. 

369.  An  important  method  for  expanding  expressions 
into  series  is  based  on  the  following  theorem  : 

The  Theorem  of  Undetermined  Coefficients. 

If  the  series  A  -\-  Bx  -\-  Cx'  -h  Bx^  +  •••  is  always  equal  to 
the  series  A'  -}-  B'x  +  G'x^  +  D'x^  +  •••>  ivhen  x  has  any  value 
which  makes  both  series  convergent,  the  coefficients  of  like 
powers  of  x  in  the  series  will  he  equal;  that  is,  A  =  A', 
B  =  B',  G=C',etc. 

For  since  the  equation 

A  -^  Bx  +  Cx"  -\-  Dx^  -i-  '•'  =  A'  -{-  B'x  -^  C'x^  -f  D'x^  +  ••. 

is  satisfied  when  x  has  any  value  which  makes  both  mem- 
bers convergent,  and  since  both  members  are  convergent 
when  ic  =  0  (§  368),  it  follows  that  the  equation  is  satisfied 
when  a;  =  0. 

Putting  x  =  0,  we  have  A  =  A'. 

Subtracting  A  from  the  first  member  of  the  equation,  and 
its  equal  A'  from  the  second  member,  we  obtain 

Bx  +  Cx^  -\-Dx'-^'"  =  B'x-\-  Cx"  -{-  D'x^  +  .... 

Dividing  each  term  by  x, 

B-^Cx-\-Dx'-\-'"  =  B'  +  Cx  +  D'x"  +  •••• 

This  equation  also  is  satisfied  when  x  has  any  value 
which  makes  both  members  convergent ;  and  putting  x  =  0, 
we  have  B  =  B'. 

In  like  manner,  we  iliay  prove  C  =  C,  D  =  D',  etc. 

370.  A  finite  series  being  always  convergent,  it  follows 
from  the  preceding  article  that  if  two  finite  series 

A-]-Bx-j-Cx'+  ...  +AV  and  A'  +  B'x-^C'x--{-  .••  +K'x^ 

are  equal  for  every  value  of  x,  the  coefficients  of  like  powers 
of  X  in  the  two  series  are  equal. 


UNDETERMINED   COEFFICIENTS.  321 


EXPANSION  OF  FRACTIONS  INTO  SERIES. 


2-;u 


,2  _  a^ 


371.   1.    Expand -^ '—  in  ascending  powers  of  x. 

1  —  2x  -\-  Sx- 

Assume  ^--Sg^-x^  ^  ^  ^  ^Jx  +  Cx^  +  Dx^  +  ^x*  +  ••• ;         (1) 
1  -  2  a;  +  3  x"-^ 

where  A,  B,  C,  D,  £*,  etc. ,  are  quantities  independeut  of  x. 

Clearing  of  fractions,  and  collecting  the  terms  in  the  second  mem- 
ber involving  like  powers  of  x,  we  have 


2-Zx^-x»  =  A+    B 
-2  A 


x-\-  C 
-2B 
+  3^ 


x2+  D 
-2C 
+  35 


x3+     E 
-2D 

4-3C 


x*+....     (2) 


The  second  member  of  (1)  must  express  the  value  of  the  fraction 
for  every  value  of  x  which  makes  the  series  convergent  (§  307). 

Hence,  equation  (2)  is  satisfied  when  x  has  any  value  which  makes 
both  members  convergent ;  and  by  the  Theorem  of  Undetermined 
Coefficients,  the  coefficients  of  like  powers  of  x  in  the  series  are  equal. 

Then,  A  =  2. 

B-2A  =  0;      whence,  B  =  2A  =4. 

C-2B4-3.4=-3;  whence,  C  =  2B-3^-3=-l. 

D-2C+SB  =  -l;  whence,  D  =  2C-SB-\=-U. 

E-2D  +  SC  =  0;      whence,  E  =  2D-SC        =-27;  etc. 
Substituting  these  values  in  (1),  we  have 

2-3a;2-a^  =2  +  4a;  -  a;^  -  ISar^  -  27  a:^  +  ...,     Ans. 
l-2a;  +  3a;2 

The  result  may  be  verified  by  division. 

Note  1.  A  vertical  line,  called  a  bar,  is  often  used  in  place  of  a 
parenthesis. 

Thus,  +    B  \xis  equivalent  to  (B  —  2  A)x. 

-2a\ 

Note  2.  The  result  expresses  the  value  of  the  given  fraction  only 
for  such  values  of  x  as  make  the  series  convergent  (§  367). 

If  the  numerator  and  denominator  contain  only  even 
powers  of  x,  the  operation  may  be  abridged  by  assuming  a 
series  containing  only  the  even  powers  of  x. 


322  ALGEBRA. 

2  4-  4:x'^  —  X* 
Thus,  if  the  fraction  were  — — ,  we  should  as- 

sume  it  equal  to  ^  +  Bx^  +  Cx'^  +  Dx^  +  j&V  +  •  •  •. 

In  like  manner,  if  the  numerator  contains  only  odd 
powers  of  x,  and  the  denominator  only  even  powers,  we 
should  assume  a  series  containing  only  the  odd  powers  of  x. 

If  every  term  of  the  numerator  contains  x,  we  may  as- 
sume a  series  commencing  with  the  lowest  power  of  x  in 
the  numerator. 

If  every  term  of  the  denominator  contains  x,  we  determine 
by  actual  division  what  power  of  x  will  occur  in  the  first 
term  of  the  expansion,  and  then  assume  the  fraction  equal 
to  a  series  commencing  with  this  power  of  x,  the  exponents 
of  X  in  the  succeeding  terms  increasing  by  unity  as  before. 


2.   Expand  — -^ — —  in  ascending  powers  of  x. 

O  3/    —  X^ 

Dividing  1  by  3  x^,  the  quotient  is  — ;  we  then  assume 

^        =  Ax-^  +  Bx-^  +  C-\-  Dx  +  Ex'^  +  ....  (1) 


3  a;2  -  X 
Clearing  of  fractions,  we  have 

l  =  3^  +  3J5|a;  +  3C|a;2  +  3Z)|x3  +  3J5;|a;4+  .... 

-^I-^I     -    c\    -  d\ 

Equating  the  coefficients  of  like  powers  of  «, 

3^  =  1. 
35-^  =  0. 
3  O  -  J5  =  0. 
32)-  0  =  0. 
3  j^  -  D  =  0 ;  etc. 

Whence,        A  =  \  B  =  -,  C  =  —,  D=—,  E  =  —,  etc. 
3'  9'  27'  81'  243' 

Substituting  in  (1),  we  have 


V  +  ^  +  -^  +  £-  +  ^+  •••,  Ans. 


3  x2  -  a;8       3        9       27     8i      243 


UNDETERMINED   COEFFICIENTS. 


323 


EXAMPLES. 

Expand  each  of  the  following  to  five  terms  in  ascending 
powers  of  x : 
3    1  -f  5a;  o     2a;  +  3r^ 

1+a;  ' 
.    3-2a; 
■    1-A.x 
-    2-f  7a;^ 

-a? 


6. 


7. 


2-h3a;2 

1-x- 
\-2x-x' 


3ar^ 


8. 


9. 


10. 


11. 


12. 


iJ^hx-lx"' 

1 

^x'-bx^' 
l-2a; 
2-3x-\-4:x'' 

l-4ar^  +  6a;^ 

1  +  2a;-ic2  • 

2  +  a;-3a;^ 
1  —  4:X-\-  oay^ 


13. 


14. 


15. 


16. 


17. 


1   —    7  X^  —  4:X^ 

x'-5x*-2a^' 

x^-Sa^-^-x*' 

a^-4:X^  +  2a^ 
2-Sx'-x^' 

3-2x-\-a/ 
3-4aT^ 

2a;4-ar'-3iB*' 


EXPANSION  OF  RADICALS  INTO  SERIES. 


372.   1.   Expand  Vl  —  a;  in  ascending  powers  of  x. 
Assume  Vl  -  x  =  A  ^  Bx  +  Cx^  +  Dx^  +  ^x*  + .... 
Squaring  both  members,  we  have  by  the  rule  of  §  187, 


(1) 


1  -  X  =  ^2  I  a;  +       ^ 

^2Ab\     -\-2AC 


X2 


-^2  AD 
+  2BC 

Equating  the  coefficients  of  like  powers  of  x, 
^•^  =  1 ;      whence,  ^  =  1. 
2AB^-\ 


X8+         C2|X*+.. 

^2Ae\ 
+  2Bd\ 


whence,  B  = = 

2A         2 


52  +  2^C  =  0 

2^2)  +  25C=0 

C^^2AE-¥2BD  =  0 


whence,  C 


2A 


TiC 

whence,  D  = = 

A 


whence,  E  =  — 


1 
8* 
J_ 
16' 
C^-{-2BD 


Substituting  these  values  in  (1),  we  have 
x2      x3     5x* 


\/n^  =  i 


8      16      128 
The  result  may  be  verified  by  evolution. 


2A 


,  Ans. 


128 


etc. 


324  ALGEBRA. 


EXAMPLES. 


Expand  each  of  the  following  to  five  terms  in  ascending 
powers  of  x : 


2.    Vl+4a;.  4.    Vl-f2a;-a^.      6.    </l+^x. 


3.    Vl-5a;.  5.    Vl-a^-a^.         7.    Vl-a^  +  a.-^. 

PARTIAL  FRACTIONS. 

373.  If  the  denominator  of  a  fraction  can  be  resolved 
into  factors,  each  of  the  first  degree  in  x,  and  the  numerator 
is  of  a  lower  degree  than  the  denominator,  the  Theorem  of 
Undetermined  Coefficients  enables  us  to  express  the  given 
fraction  as  the  sum  of  two  or  more  partial  fractions,  whose 
denominators  are  factors  of  the  given  denominator,  and 
whose  numerators  are  independent  of  x. 

374.  Case  I.     When  no  tivo  factors  of  the  denominator 

are  equal. 

lOic  4-  1 
1.    Separate ^ —  into  partial  fractions. 

(Zx-l){bx  +  2) 

Assume  l^^  +  i .  _^  +  _A_,  (i) 

where  A  and  B  are  quantities  independent  of  x. 

Clearing  of  fractions,       19 a;  +  1  =  ^(5  x  +  2)  +  ^(3  x  -  1). 

Or,  \^x-l\=(jSA+ZB)x  +  2A- B.        (2) 

The  second  member  of  (1)  must  express  the  value  of  the  given  frac- 
tion for  every  value  of  x. 

Hence,  equation  (2)  is  satisfied  by  every  value  of  x ;  and  by  §  370, 
the  coefficients  of  like  powers  of  x  in  the  two  members  are  equal. 

That  is,  5  ^  +  3  5  =  19, 

and  2A-     B=l. 

Solving  these  equations,  we  obtain  A  =  2  and  ^  =  3. 
Substituting  in  (1),    19^+_i ^  _2 —  _^  — 3 —  ^^^ 

The  result  may  be  verified  by  adding  the  partial  fractions. 


UNDETERMINED  COEFFICIENTS.  325 

2.    Separate —^ into  partial  fractions. 

2  a;  —  ar  —  ar 

The  factors  of  2x  —  x^  —  x^  are  x,  I  ~  x,  and  2  +  x  (§  284). 

Assume  then        ^  "^  ^       =  ^  +  — ^  +  — ^. 
2a;-ic2-ic3      x      l-x-2  +  x 

Clearing  of  fractions,  we  have 

x  +  i  =  A{l-x){2-\-x)-h  Bx{2  +  a;)  +  Cx{\  -  x). 

This  equation  is  satisfied  by  every  value  oi  x\  it  is  therefore  satis- 
fied when  a;  =  0. 

Tutting  X  =  0,  we  have  4  =  2  A,  or  A  =  2. 
Again,  the  equation  is  satisfied  when  x  =  1. 
Putting  X  =  1,  we  have  5  =  3  jB,  or  B  =  -- 

o 


The  equation  is  also  satisfied  when  x  =  —  2. 
Putting  X  =  —  2,  we  have  2  =  —  6  C,  or  C  = 


_1 
3* 
5  1 


x+4        _2         3  3 

^'''  2x-x2-x8~x  +  rr^"^2T^ 

^x"^3(l-x)~3(2  +  x)'  ^^^*' 

Note.  To  find  the  value  of  A,  in  Ex.  2,  we  give  to  x  swcA  a  vahie 
as  will  make  the  coefficients  of  B  and  C  equal  to  zero ;  and  we  proceed 
in  a  similar  manner  to  find  the  values  of  B  and  C. 

This  method  of  finding  ^,  B,  and  C  is  usually  shorter  than  that 
used  in  Ex.  1. 

EXAMPLES. 
Separate  each  of  the  following  into  partial  fractions : 
a;2-75  „        ax-l^a" 


18  a; 

+  3 

4ar» 

-9 

X- 

-2 

6. 


a:^  — 25  a;  a^^-f  4aa;— 5a^ 


6         38 a; +  5  g         46 -5a; 


5ar^-6a;  6a;2  +  5a'-6  8-18a;-5ar^ 

a.-^  4- 10a; -7  ^^     -  13a;^  + 27a;  +  18 

(2a;-l)(12ar^-a;-6)*  '     (ar^  -  2  a;)  (x^  -  9)  * 


326  ALGEBRA. 

375-  Case  II.  When  all  the  factoids  of  the  denominator 
are  equal. 

Let  it  be  required  to  separate  -— ^~ —  into  partial 

fractions.  ^  ^ 

Substituting  y  -\-S  for  x,  the  fraction  becomes 

(y  -\.3y  -ll(y  -{-  3)  +  26  ^f  -  5y  +  2  ^1      5  ^  2 

f  y^  y    y^    f 

Eeplacing  y  hj  x  —  3,  the  result  takes  the  form 

1  5.2 


x-3     (x-3y     (x  -  3y 

This  shows  that  the  given  fraction  can  be  expressed  as 
the  sum  of  three  partial  fractions,  whose  numerators  are 
independent  of  x,  and  whose  denominators  are  the  powers 
of  aj  —  3  beginning  with  the  first  and  ending  with  the  third. 

Similar  considerations  hold  with  respect  to  any  example 
under  Case  11. ;  the  number  of  partial  fractions  in  any  case 
being  the  same  as  the  number  of  equal  factors  in  the 
denominator  of  the  given  fraction. 

EXAMPLES. 


6a^+5    ,, 

(3a;  +  5y 


376.   1.  Separate  ^^  _~;  ^  ^  ^^*^  partial  fractions. 


In  accordance  with  the  principle  stated  in  §  375,  we  assume  the 
given  fraction  equal  to  the  sum  of  two  partial  fractions,  whose  denomi- 
nators are  the  powers  of  3  x  +  5  beginning  with  the  first  and  ending 
with  the  second. 

Thus,  _6^+_5_=,^_  + B 

(3x  +  6)2     3a; +  5      (3x  +  5)2 

Clearing  of  fractions,    6  x  +  5  =  ^(3  x  +  5)  +  -B. 

Or,  6x  +  5  =  3^x  +  5^-|-5. 

Equating  the  coeflacients  of  like  powers  of  x, 

3^=6. 

5A  +  B  =  b. 


UNDETERMINED   COEFFICIENTS.  327 

Solving  these  equations,  we  have  A —  2  and  ^  =  —  5. 

Whence,  ' == » — •»  Ans. 

'  (3x  +  5)2     3x  +  5      (3x  +  5)2' 

Separate  each  of  the  following  into  partial  fractions: 

14a; -30  .     9a;^-15a;-l      g    10ar^  +  3a;-l 

4a;2_i2a;  +  9*       '        {3x-iy     '       '        (5x-h2y 

3    x'-\-4:X-l  5     8  ar^- 19  .^    a:^-3a;^-a; 

*     (x-{-5y  '       '  {2x-3y  '     (x-iy 

g    a^-\-4:x'-\-7x-\-2  g    18a;^-21g^  +  4a; 

(x-^2y         '  '  {Sx-2y 

377.  Case  III.  When  some  of  the  factors  of  the  denomi- 
nator are  equal. 

1.   Separate  —^ — ^rV  ^^^^  partial  fractions. 

X\X  -\-\) 

The  method  in  Case  III.  is  a  combination  of  those  of  Cases  I.  and  II. 

x2-4ic  +  3     A  ,      B     ,         C 

We  assume  — ; -^  =  —  +  — — :  +  ; — — ^r^- 

x{x  +  1)2        X      x-\-\      (X  +  1)2 

Clearing  of  fractions,  x2  _  4  a;  +  3  =  ^(x  +  1)2  +  Bx{x  +  1)  +  Cx 

=  (^+5)x2  +  (2^  +  5+C)x+A 

Equating  the  coefficients  of  like  powers  of  x, 

^  +  ^  =  1. 

2^  +  5+C  =  -4. 

^  =  3. 

Solving  these  equations,  we  have  ^  =  3,  ^B  =  —  2,  and  C  =  —  8. 

Whence,  x^zzi£±3  ^3 2 8  ^^^ 

'  ;c(x+l)2       X     x  +  1      (x  +  l)2' 

Note.  It  is  impracticable  to  give  an  illustrative  example  for  every 
possible  case  ;  but  no  difficulty  will  be  found  in  assuming  the  proper 
partial  fractions  if  attention  is  given  to  the  following  general  rule. 


328  ALGEBRA. 

The  fraction should  be  put  equal  to 

(x  +  a){x  +  6)  •••  {X  +  my  ••• 


X  +  «     x-\-h  x  +  m     {x  +  m)2  (oj  +  m)'' 

Single  factors  like  x  +  a  and  a;  +  ?)  having  single  partial  fractions 
corresponding,  arranged  as  in  Case  I. ;  and  repeated  factors  like 
{x  +  my  having  r  partial  fractions  corresponding,  arranged  as  in 
Case  II. 

EXAMPLES. 
Separate  each  of  the  following  into  partial  fractions : 


xix-^y 


5. 


3    8a^  +  8a^-18fl;-8  g 

a;*  +  4  ic^ 


12aj2-lla;-38 


7. 


2- 

_3a;_a^_2a^ 

0^(0.-1)^ 

4 

-9aj-12a^-2a^ 

0,(0^  +  1)  (a. +  2)2 

3aj  +  13 

(3aj-l)(2a;  +  3)2  (2  a;  -  3)(8a^  -  10  a;  -  3) 

378.  If  the  degree  of  the  numerator  is  equal  to,  or  greater 
than,  that  of  the  denominator,  the  preceding  methods  are 
inapplicable. 

In  such  a  case,  we  divide  the  numerator  by  the  denomi- 
nator until  a  remainder  is  obtained  which  is  of  a  lower 
degree  than  the  denominator. 

/pS  _  3ic2  _  ^ 

1.    Separate into  an  integral  expression  and 

of  —  X 

partial  fractions. 

Dividing  x^  —  3  ai^  —  1  by  cc^  _  aj^  the  quotient  is  x  —  2,  and  the 
remainder  —  2  a;  —  1. 

Then,  »^-S^'-l  =  x-2+-f»-l. 

X^  —  X  X^  —  X 

Separating  — — ^  ~     into  partial  fractions  by  the  method  of  Case 

x^  —  X 

1  •'       3 

I.,  the  result  is  — ■ 


X     x  —  \ 
Whence,        a;8-3a;2-l  ^  ^  _  g  +  1 3    ^  ^^s. 

X^  —  X  X      x—1 


UNDETERlVnNED  COEFFICIENTS.  329 


EXAMPLES. 

Separate  each  of  the  following  into  an  integral  expression 
and  two  or  more  partial  fractions : 

2  9ar^4-9a^-6  ^    x' +  2x^  -  3x^ -\- x -\-3 
(«-f  2)(3ir-l)*  '  x'ix-^l) 

3  2a^-17x'  +  Ux-29        g    ar^  -  2a^  +  4a;  -  1 

(x-2y         '      ■       a^(x-iy 

g    ixf  +  Sx^-^3x'-10a^-x-{-6 

x*  +  3x^ 

379.  If  the  denominator  of  a  fraction  can  be  resolved 
into  factors  partly  of  the  first  and  partly  of  the  second 
degree,  or  all  of  the  second  degree,  in  x,  and  the  numerator 
is  of  a  lower  degree  than  the  denominator,  the  Theorem  of 
Undetermined  Coefficients  enables  us  to  express  the  given 
fraction  as  the  sum  of  two  or  more  partial  fractions,  whose 
denominators  are  factors  of  the  given  denominator,  and 
whose  numerators  are  independent  of  x  in  the  case  of 
fractions  corresponding  to  factors  of  the  first  degree,  and 
of  the  form  Ax  +  jB  in  the  case  of  fractions  corresponding 
to  factors  of  the  second  degree. 

1.    Separate  — — -  into  partial  fractions. 

The  factors  of  the  denominator  are  x  +  1  and  x^  —  x  +  1 . 

Assume  then  — ? —  =  -^  +    Bx  +  C  ,j. 

x^+l      x+  I      x'^-x+1  ^  '^ 

Clearing  of  fractions,     I  =  A(x^  -  x  i- l)  +  (Bx+  (7)(x4-l). 
Or,  l=(A  +  B)x^-\-(-A  +  B-^  C)x  +  A+  a 

Equating  the  coefficients  of  like  powers  of  x, 

A  +  B  =  0. 
'-A+B-\-C  =  0. 

A-{-C=l. 


330 


ALGEBRA, 


Solving  these  equations,  we  have 


A  =  -,  5  =  --,  and  (7  =  -- 
3  3  3 


Substituting  in  (1), 


x-2 


x3+l     3(x+l)      3(x2-x+l) 


,  Ans. 


^-  ^^^1 


EXAMPLES. 
Separate  each  of  the  following  into  partial  fractions : 

3          a;^  4- 16  a;  - 12  «  12+13a;-2a^ 

(Sx  +  l)(x'-x-\-3)'  '  8ar^-27 

^      ^x'  +  llx-l  »  2a^  +  2a.-^  +  10 

(2x-5)(x'-i-2)'  "  a;4_f_a^_^l 


REVERSION  OF  SERIES. 

380.   To  revert  a  given  series  y  =  a  -f-  bx""  -\-  ex"  -\ is  to 

express  x  in  the  form  of  a  series  proceeding  in  ascending 
powers  of  y. 

1.   Revert  the  series  y  =  2x-\-a^  —  2x^  —  Sx*-\-  •••. 

Assume  x  =  Ay  -\-  By^  +  Cy^-{-  Dy^  +  •••.  (1) 

Substituting  in  this  the  given  value  of  y,  and  performing  the  opera- 
tions indicated,  we  have 

a;  =  ^(2x  +  x2-2x3-3x4  +  •••) 

+  5(4  x2  +  X*  +  4  a:3  -  8  x4  +  -..) 
+  0(8x3  + 12  x4  +  ...)+Z)(16a;4+  •••)+•••. 
x4  +  .... 


That  is,    X  =  2  ^x  +    A 

x2-2^ 

x3-    3^ 

+  45 

+  45 

-    75 

+  80 

+  12  0 
+  16  5 

Equating  the  coefficients  of  like  powers  of  x, 

2^  =  1. 

^  +  45  =  0. 

-2^  +  45  +  80  =  0. 

-3^-75  +  12 

0+16  2> 

=  0;  etc. 

UNDETERMINED   COEFFICIENTS.  331 

Solving  these  equations, 


A  = 

i^  = 

1 
8' 

C  = 

3 

,  D  = 

13 

128' 

,  etc. 

Substituting  in 

.  (1),  X-. 

^h 

8^   ^ 

A  ,3. 

128^^ 

Ans. 

If  the  even  powers  of  x  are  wanting  in  the  given  series, 
the  operation  may  be  abridged  by  assuming  x  equal  to  a 
series  containing  only  the  odd  powers  of  y. 

EXAMPLES. 
Revert  each  of  the  following  to  four  terms : 

2.    y  =  X  —  x^  +  x^  —  X*  -\ . 

^  2      3      4 

4.  y  =  x-\-2x^-^Sa^-\-4:X*-{ . 

5.  y  =  x  —  Sx^-\-5x^— 7  X* -{-"-. 

a»2         /j«3         ^4 

7.  2,  =  ?  +  ?'  +  ^  +  ?'+.... 
*     2     4      6      8 

8.  y  =  x  +  3?  +  2x?-^Bx'-\ . 

9.  y  =  x+=f  +  t  +  =i+.... 

357 


332  ALGEBRA. 

XXXIII.    THE  BINOMIAL  THEOREM. 

FRACTIONAL  AND  NEGATIVE  EXPONENTS. 

381.  It  was  proved  in  §  359  that,  if  n  is  a  positive  integer, 
(l  +  ^)"  =  l  +  n^+^^^^:t^  +  "("-|^("-^V  +  -.(l) 

I 

382.  Proof  of  the  Theorem  for  a  Fractional  or  Negative 
Exponent. 

I.    When  the  exponent  is  a  positive  fraction. 

Let  the  exponent  be  -,  where  p  and  q  are  positive  integers. 

p     ^  \ 

By  §  211,  (1  +  x).  =  -V(l  +  xy  =  -^l+pa^+...,  by  (1). 

It  is  evident  that  a  process  may  be  found,  analogous  to 
those  of  §§  194  and  200,  for  expanding  ■\/' 1 -i- px -\- • --  in 
ascending  powers  of  x ;  and  the  first  term  of  the  result  will 
evidently  be  1. 

Assume  then,  ^/l -\-px -]- -"  =  1 -\- Mx-\- JVx^ -^  -".       (2) 
Raising  both  members  to  the  gth  power,  we  have 

1  +^^_  ...  =  [1  ^.(Mx  +  iVa^-h  ...)? 

=  1  -^q(Mx  +  iV^a^  +  •••)+  '",  by  (1). 

This  equation  is  satisfied  by  every  value  of  x  which  makes 
both  members  convergent;  and  by  the  Theorem  of  Unde- 
termined Coefficients  (§  369),  the  coefficients  of  x  in  the  two 
series  are  equal. 

P 
That  is,  p  =  qM,  or  3f  = -• 

Substituting  this  value  in  (2),  we  have 

(l-faj>-=l+|aj+—  (3) 


THE   BINOMIAL   THEOREM.  333 

II.  When  the  exponent  is  a  negative  integer  or  a  negative 
fraction. 

Let  the  exponent  be  —  s,  where  s  is  a  positive  integer  or 
a  positive  fraction. 

By  §  214,  (1  +  .)-.  =  ^.  =  ^^_,  by  (1)  or  (3). 

It  is  evident  that can  be  expanded  by  division 

in  a  series  proceeding  in  ascending  powers  of  x-,  thus, 

1  +  S«-|-  '•')'^(X  —SX+  '" 

lH-sa;-f--'- 

—  SX—  '" 

That  is,  (1  +  it')"  =  1  -  SX  -j-  . . ..  (4) 

From  (3)  and  (4),  we  observe  that,  when  n  is  fractional 
or  negative,  the  form  of  the  expansion  is 

(1  -h  a;)"  =  1  4-  nx  -{- Ax^ -\- Bx^  +  -".  (5) 

Writing  -  in  place  of  x,  we  obtain 
a 


M 


X       .x^      ^a? 


Multiplying  both  members  by  a", 

(a  +  xy  =  a"  +  na'^-^x  +  ^a'*- V  -f  Ba^'-^a?  H .       (6) 

This  result  is  in  accordance  with  the  second,  third,  and 
fourth  laws  of  §  357 ;  hence,  these  three  laws  hold  for  frac- 
tional or  negative  values  of  the  exponent. 

We  will  now  prove  that  the  fifth  law  of  §  357  holds  for 
fractional  or  negative  values  of  the  exponent. 

Let  P  denote  the  coefficient  of  a;*",  and  Q  the  coefficient  of 
ic*'+^,  in  the  second  member  of  (5). 

Then  (5)  and  (6)  may  be  written 

(1  +  a;)"  =  1  4-  na;  -f  •••  +  iV-  +  Qaf+i  +  •••,  (7) 

and  (a  -f  xy  =  a"  +  na'^^x  -}-••• 

+  Pa'^-'x'-  +  Qa'»-'- V+^  +  •  • ..  (8) 


334  ALGEBRA. 

In  (8)  put  a  =  1  +  ?/  and  x  =  z;  then, 

(1  +  2/  +  2=)"  =  (1  +  VT  +  •••  +  P(l  +  2/)"-'-2!'-  +  ..-.       (9) 
Again,  in  (7)  put  x  =  z-\-y'^  then, 

(1  +  z  +  yy  =  1  +  '"  +  P{z  +  yy  +  q(z  ^  yy^^ 4-  •••. 
Expanding  the  powers  oi  z  -\-yhj  aid  of  (8),  we  have 

(1  +  2;  +  y)'*  =  1  +  ...  +  P[z^  +  rz'-^y  +  •••] 

4- e[^''+^  +  (r  +  lK2/ +...]  +  ....   (10) 

Since  the  first  members  of  (9)  and  (10)  are  identical,  their 
second  members  must  be  equal  for  every  value  of  z  which 
makes  both  series  convergent;  and  by  the  theorem  of  Unde- 
termined Coefficients,  the  coefficients  of  2'"  in  the  two  series 
are  equal. 

Or,  P(l  4-  yy-'  =  P+  Q{r  +  l)y-\-  terms  in  y"^,  f,  etc. 

Expanding  the  first  member  by  aid  of  (7),  this  becomes 

P[l  4-  {n  -  r)y  4-  ••.]  =  P  +  Qir  +  %  +•... 

This  equation  is  satisfied  by  every  value  of  y  which  makes 
both  members  convergent,  .and  hence  the  coefficients  of  y  in 
the  two  series  are  equal. 

That  is,  P{n  -  r)  =  Q(r  +  1),  or  Q  =  P(n-r)^ 

r-\-l 

But  in  the  second  member  of  (8),  n  —  r  is  the  exponent 
of  a  in  the  term  whose  coefficient  is  P,  and  r  4- 1  is  the 
exponent  of  x  in  that  term  increased  by  1. 

Hence,  the  fifth  law  of  §  357  has  been  proved  to  hold  for 
fractional  or  negative  values  of  the  exponent. 

By  aid  of  the  fifth  law,  the  coefficients  of  the  successive 
terms  after  the  second,  in  the  second  member  of  (8),  may 
be  readily  found  as  in  §  358 ;  thus, 

(a  +  xy  =  a"  -h  na^-^x  4-  ^^^  ~  ^^  a"  V 

n(n-l)(n_--2}^,.3^      ^..^       ^^^^ 


THE   BINOMIAL   THEOREM.  335 

The  second  member  of  (11)  is  an  infinite  series ;  for  if  n 
is  fractional  or  negative,  no  one  of  the  quantities  n  —  1, 
II  —  2,  etc.,  can  become  equal  to  zero. 

The  result  expresses  the  value  of  (a  -\-  x)"  only  for  such 
values  of  a  and  x  as  make  the  series  convergent  (§  367). 

EXAMPLES. 

383.  In  expanding  expressions  by  the  Binomial  Theorem 
when  the  exponent  is  fractional  or  negative,  the  exponents 
and  coefficients  of  the  terms  may  be  obtained  by  aid  of  the 
laws  of  §  357,  which  have  been  proved  to  hold  universally. 

If  the  second  term  of  the  binomial  is  negative,  it  should 
be  enclosed,  sign  and  all,  in  a  parenthesis  before  applying 
the  laws ;  if  either  term  has  a  coefficient  or  exponent  other 
than  unity,  it  should  be  enclosed  in  a  parenthesis  before 
applying  the  laws. 

1.  Expand  (a  +  x)^  to  four  terms. 

The  exponent  of  a  in  the  first  term  is  f  ;  in  the  second  term,  —  ^  ;  in 
the  third  term,  —  | ;  in  the  fourth  term,  —  |  ;  etc. 

The  exponent  of  x  in  the  second  term  is  1  ;  in  the  third  term,  2  ; 
in  the  fourth  term,  3  ;  etc. 

The  coefficient  of  the  first  term  is  1  ;  of  the  second  term,  f  ;  multi- 
plying the  coefficient  of  the  second  term,  |,  by  —  |,  the  exponent  of 
a  in  that  term,  and  dividing  the  result  by  the  exponent  of  x  in  the 
term  increased  by  1,  or  2,  we  have  —  ^  as  the  coefficient  of  the  third 
term  ;  and  so  on. 

Then,  (a  +  x)^  =  a^  +  f  a'^x  -  I a'^x'^  +  j\ a'^x^ ,  Ans. 

2.  Expand  (1  —  2x~^-)-^  to  five  terms. 

(l-2a;~5)-2  =[!+(_  2.x-^)]-1i 

=  1-2  -2.1-3.  (._  2a;"^)+  3  •  l"*  •  (-2x~h^ 

-  4  .  1-5  .  (-  2a;"2)3  +  5  •  l-«  •  (-  2x~h*  -  ... 
=  1  +  4  x~^  -f  12  x-^  +  32  x""2  -f  80 x-^  +  •  ••,  Ans. 


336                                      ALGEBRA. 
3.   Expand to  five  terms. 

1  1  =[(«-!)  + (3  x^)]-^ 


^(a-i  +  Sx^)      (a-i  + 3x3)3 


.4  1  _1 


-  if  (a-i)"'^(3xb^  +  3j¥ir  ia-^y'^\sx^y  -  - 

1  4    1  7     2  10  lA    4 


Expand  each  of  the  following  to  five  terms : 

V 


4. 

(a-^x)^. 

5. 

{l  +  x)-\ 

6. 

(i-x)-i 

7. 

Va  —  X. 

fi 

1 

S/l+^ 

9. 

1 

10.    {xi-2y)i.   ^15. 


.1 


11.    (m-2+vV'.    ,         ,        ^,_3 

y  12.    (a'-2x^^yK 

1  .  17.  ^[(a.-t-32/t)-]. 

y  .        18.     ^-</^)' 

(a  -  by       V  14.    (aj-3  +  22/^)^.  V^  / 

384.  The  formula  for  the  rth  term  of  (a  +  xy  (§  361) 
holds  for  all  values  of  n,  since  it  was  derived  from  an 
expansion  which  has  been  proved  to  hold  universally. 

EXAMPLES. 

.      1.   Find  the  7th  term  of  {a  -  3a;"^)"i 

We  have,  (a  -  3  x~h~^  =  [a  +  ( -  3  x~^) ]"i 

In  this  case,  n  —  —  \  and  r  =  7. 
The  exponent  of  (-  Zx^  is  7  -  1,  or  6. 
The  exponent  of  a  is  —  ^  -  6,  or  —  ^. 

The  first  factor  of  the  numerator  is  —  \,  and  the  last  factor 
_>  i_9  +  1,  or  -  Y-. 

The  last  factor  of  tjie  denominator  is  6. 


THE   BINOMIAL   THEOREM. 


337 


Hence,  the  7th  term 

1     _4     _7         10         13 
3  "      3  '       3  '       3   *       3 


1.2.3.4 


16 

3       _19  _3 


Note.     The  note  to  §  362  applies  with  equal  force  to  the  examples 
in  the  present  article. 


Find  the 

2.  5th  term  of  (a  +  x)K 

3.  7th  term  of  (a  +  &)"^. 

4.  12th  term  of  (1  -  x)-\ 
^'  5.   6th  term  of  {x-^^-^y^y. 

.     6.   9th  term  of  {(i-^2xf. 


7.   5th  term  of 


V(l  -  xf 
8.   7th  term  of  (a'  -  x^)^. 
1 


9.   10th  term  of 


{x  -\-  mf 


10.  8th  term  of  (m*-  2  ?i-^)"l 

11.  9th  term  of  V(a  -  x)\ 

12.  6th  term  of  (a^  -  fe-^)"^. 

13.  8th  term  of  {x-^  -h  S?/"^)"^. 

14.  10th  term  of  (x  Vf  -  -r^V** 

15.  11th  term  of  (a^  +  3  6"*)i 

385.  Extraction  of  Roots  by  the  Binomial  Theorem. 

1.   Find  V25  approximately  to  live  places  of  decimals. 
We  have,  v^  =  25^  =  (27  -  2)^  =  (3*  -  2)i 

Expanding  by  the  Binomial  Theorem,  we  have 

[(38)  +  (-2)p=(38)U|(38)-^(-2)-l(38)-f(-2)2 


338  ALGEBRA. 


Or,  .   ^^  =  3--^-  + 


40 


3.32     9.35    .81.38 

Expressing  the  value  of  each  fraction  approximately  to  the  nearest 
fifth  decimal  place,  we  have 

V'25  =  3  -  .07407  -  .00183  -  .00008 =  2.92402  ... ,  Ans. 

Rule. 

Separate  the  given  number  into  two  parts,  the  first  of  ivhich 
is  the  nearest  perfect  poiver  of  the  same  degree  as  the  required 
root. 

Expand  the  result  by  the  Binomial  Theorem. 

Note.  If  the  second  term  of  the  binomial  is  small  compared  with 
the  first,  the  terms  of  the  expansion  diminish  rapidly  ;  but  if  the 
second  term  is  large  compared  with  the  first,  it  requires  a  great  many 
terms  to  ensure  any  degree  of  accuracy. 

EXAMPLES. 

'  Find  the  approximate  value  of  each  of  the  following 
to  five  places  of  decimals: 


2.    V26. 

4.    </9. 

6.    ^17. 

3.    V98. 

6.    ^/78. 

7.    </29. 

LOGARITHMS.  339 


XXXIV.    LOGARITHMS. 

386.  Every  positive  number  may  be  expressed,-  exactly 
or  approximately,  as  a  power  of  10. 

Thus,  100  =  102 ;  13  =  10i"39... .  etc. 

When  thus  expressed,  the  corresponding  exponent  is 
called  its  Logarithm  to  the  Base  10. 

Thus,  2  is  the  logarithm  of  100  to  the  base  10 ;  a  relation 
which  is  written  log^olOO  =  2,  or  simply  log  100  =  2. 

387.  Logarithms  of .  numbers  to  the  base  10  are  called 
Common  Logarithms,  and,  collectively,  form  the  Common 
System. 

They  are  the  only  ones  used  for  numerical  computations. 

Any  positive  number,  except  unity,  may  be  taken  as  the 
base  of  a  system  of  logarithms ;  thus,  if  a'  =  m,  where 
a  and  m  are  positive  numbers,  then  x  =  log«  m. 

Note.    A  negative  number  is  not  considered  as  having  a  logarithm. 


38a   By  §§  213  and  214, 

W  =  1, 

io-=i=.i, 

10^  =  10, 

io-=i;^=oi, 

102  =  100, 

l^"  =  l^=-^01,eta. 

Whence  by  the  definition  of  §  386, 

log  1  =  0, 

log.l  =  -l  =  9-10, 

log  10  =  1, 

log.01=-2  =  8-10, 

log  100  =  2, 

log  .001  =  -  3  =  7  -  10,  etc. 

Note.     The  second  form  for  log.l,  log  .01,  etc.,  is  preferable  in 
practice.     If  no  base  is  expressed,  the  base  10  is  understood. 


340  ALGEBRA. 

389.  It  is  evident  from  §  388  that  the  logarithm  of  a 
number  greater  than  1  is  positive,  and  the  logarithm  of  a 
number  between  0  and  1  negative. 

390.  If  a  number  is  not  an  fexact  power  of  10,  its  com- 
mon logarithm  can  only  be  expressed  approximately ;  the 
integral  part  of  the  logarithm  is  called  the  characteristic, 
and  the  decimal  part  the  mantissa. 

For  example,  log  13  =  1.1139. 

In  this  case,  the  characteristic  is  1,  and  the  mantissa 
.1139. 

For  reasons  which  will  apx)ear  hereafter,  only  the  man- 
tissa of  the  logarithm  is  given  in  a  table  of  logarithms  of 
numbers ;  the  characteristic  must  be  found  by  aid  of  the 
rules  of  §§  391  and  392. 

391.  It  is  evident  from  §  388  that  the  logarithm  of  a 
number  between 

1  and      10  is  equal  to  0  -f  a  decimal ; 
10  and    100  is  equal  to  1  -f  a  decimal ; 
100  and  1000  is  equal  to  2  -|-  a  decimal ;  etc. 

Therefore,  the  characteristic  of  the  logarithm  of  a  num- 
ber with  one  figure  to  the  left  of  the  decimal  point  is  0; 
with  two  figures  to  the  left  of  the  decimal  point  is  1 ;  with 
three  figures  to  the  left  of  the  decimal  point  is  2 ;  etc. 

Hence,  the  characteristic  of  the  logaiithm  of  a  number 
greater  than  1  is  1  less  than  the  number  of  places  to  the  left 
of  the  deci7nal  point. 

For  example,  the  characteristic  of  log  906328.51  is  5. 

392.  In  like  manner,  the  fogarithm  of  a  number  between 

1  and      .1  is  equal  to  9  +  a  decimal  —  10 ; 
.1  and    .01  is  equal  to  8  -f  a  decimal  —  10 ; 
.01  and  .001  is  equal  to  7  +  a  decimal  —  10 ;  etc. 


LOGARITHMS.  341 

.  Therefore,  the  characteristic  of  the  logarithm  of  a  deci- 
mal with  no  ciphers  between  its  decimal  point  and  first 
significant  figure  is  9,  with  —  10  after  the  mantissa ;  of  a 
decimal  with  one  cipher  between  its  point  and  first  signifi- 
cant figure  is  8,  with  —  10  after  the  mantissa;  of  a  decimal 
with  two  ciphers  between  its  point  and  first  significant  fig- 
ure is  7,  wdth  —  10  after  the  mantissa ;  etc. 

Hence,  to  find  the  characteristic  of  the  logarithm  of  a  num- 
ber less  than  1,  subtract  the  number  of  ciphers  between  the 
decimal  point  and  first  significant  figure  from  9,  icriting  —  10 
after  the  mantissa. 

For  example,  the  characteristic  of  log  .007023  is  7,  with 
—  10  written  after  the  mantissa. 

Note.  Some  writers  combine  the  two  portions  of  the  charac- 
teristic, and  write  the  result  as  a  negative  characteristic  before  the 
mantissa. 

Thus,  instead  of  7.6036  —  10,  the  student  will  frequently  find 
3,6036,  a  minus  sign  being  written  over  the  characteristic  to  denote 
that  it  alone  is  negative,  the  mantissa  being  always  positive. 

PROPERTIES  OF  LOGARITHMS. 

393.  In  any  system,  the  logarithm  of  1  is  0. 

For  by  §  213,  ««  =  1 ;  whence  by  §  387,  log,  1=0. 

394.  In  any  system,  the  logarithm  of  the  ba^e  is  1. 
For,  a^  =  a\  whence,  log,  a  =  1. 

395.  Ill  any  system  ivhose  base  is  greater  than  1,  the  loga- 
rithm ofO  is  —  00. 

For  if  a  is  greater  than  1,  a-=^  =  —  =  -  =  0  (§  295). 

Whence  by  §  387,  log„0  =  -x. 

Note.  No  literal  meaning  can  be  attached  to  such  'a  result  as 
loga  0  =  —  GO ;  it  must  be  interpreted  as  follows  : 

If,  in  any  system  whose  base  is  greater  than  unity,  a  number  ap- 
proaches the  limit  0,  its  logarithm  is  negative,  and  increases  without 
limit  in  absolute  value.     (Compare  Note  to  §  296.) 


342  ALGEBRA. 

396.  In  any  system,  the  logarithm  of  a  product  is  eqiial  to 
the  s(wi  of  the  logarithms  of  its  factors. 

Assume  the  equations 

a^  =  m]        ,  ,      „  ^„  _   r  X  =  log„  m, 

;  whence  by  §  387,  &«    ? 

Multiplying  the  assumed  equations, 

a"  X  a^  =  mn,  or  a^'^^  =  mn. 

Whence,      log„  mw  =  x-^y  =  log„  m  +  log„  w. 

In  like  manner,  the  theorem  may  be  proved  for  the  prod- 
uct of  three  or  more  factors. 

397.  By  aid  of  §  396,  the  logarithm  of  a  composite  num- 
ber may  be  found  when  the  logarithms  of  its  factors  are 
known. 

1.    Given  log  2  =  .3010  and  log  3  =  .4771;  find  log  72. 

log  72  =  log  (2  X  2  X  2  X  3  X  3) 

=  log2  +  log 2  +  log2  +  log3  +  log3  (§  396) 

=  3  X  log2  +  2  X  16g3  =  .9030  +   .9542  =  1.8672,  Ajis. 

EXAMPLES. 

Given  log  2  =  .3010,  log  3  =  .4771,  log  5  =  .6990,  and 
log 7  =  .8451,  find: 

2.  log  35.  7.  log  126.  12.  log  324.  17.  log  1125. 

3.  log  50.  8.  log  196.  13.  log  378.  18.  log  2625. 

4.  log  42.  9.  log  245.  14.  log  405.  19.  log  6048. 

5.  log  75.  10.  log  210.  16.  log  875.  20.  log  12005. 

6.  log  40.  11.  log  625.  16.  log  686.  21.  log  15876. 

398.  In  any  system,  the  logarithm  of  a  fraction  is  equal  to 
the  logarithm  of  the  numerator  minus  the  logarithm  of  the 
denominator. 


LOGARITHMS.  343 

Assume  the  equations 


of  =  m  , 

whence, 
a"  =  n     ■ 


a;  =  log,m, 


Dividing  the  assumed  equations, 

a?"     m  ,  ,     m 

^-»  =  — ,  or  a'-*  =  — 

Whence,      log„  ~  =  x  —  y  =  log„  m  —  log,  n. 
Ill 

399.   1.  Given  log  2  =  .3010 ;  find  log  5. 
log  5  =  log  -1^  =  log  10  -  log  2  (§  398)  =  1  -  .3010  =  .0990,  Ans. 


EXAMPLES. 

Given  log  2  =  .3010,  log  3  =  .4771,  and  log  7  =  .8451,  find : 

2.  logY-  5.  log  45.  8.  log  If.  11.  log28f 

3.  log  J.  6.  log  If.  9.  logGf  12.  log  ^. 

4.  logl4f.        7.  log  225.        10.  log  135.        13.  logllO^. 

400.  In  any  system,  the  logarithm  of  any  power  of  a  quan- 
tity is  equal  to  the  logarithm  of  the  quantity  multiplied  by  the 
exponent  of  the  power. 

Assume  the  equation  a*  =  m ;  whence,  x  =  log^  m. 
Raising  both  members  of  the  assumed   equation  to  the 
pth  power, 

a^  =  mP ;  whence,  log^  m^  z=px  —  p  log.  m. 

401.  In  any  system,  the  logarithm  of  any  root  of  a  quantity 
is  equal  to  the  logarithm  of  the  quantity  divided  by  the  index 
of  the  root. 

For,  log„  Vm  =  log„  (m^  =  -  log„  m  (§  400).  ^ 

402.  1.  Given  log  2  =  .3010 ;  find  log  2^. 

log2t=^x  log2(§400)=-x.3010=:.5017,  Ans.  I 

3  3 


344  ALGEBRA. 

Note.     To  multiply  a  logarithm  by  a  fraction,  multiply  first  by  the 
numerator,  and  divide  the  result  hj  the  (iienominator. 

2.  Given  log  3  =  .4771 ;  find  log  ^3. 

log  ^  =  l^  =  :i^.. 0596,  ^n*. 

o  o 

EXAMPLES.. 
Given  log  2  =  .3010,  log  3  =  .4771,  and  log  7  =  .8451,  find : 

3.  log3^         7.  log35^       11.  log24l  15.  log  ^l05. 

4.  log5«.         8.  log28i      12.  lo^-v/3.  16.  log -\/75. 

5.  log2l        9.  log27i      13.  log  </5.  17.  log -v^OS. 

6.  log7l      10.  logisi      14.  log^/7.  18.  log  ^l08. 
19.   Find  log  (2^  X  3^). 

By  §  396,  log  (2^  x  3?)  =  log  2  J  +  log  3?  =  i  log  2  +  |  log  3 
=  .1003  +  .5964  =  .6967,  Ans. 

Find  the  values  of  the  following : 

20.  logA/L   "22.^  log  (2^x10^).    24.  log  ^.    26.  log-^L- 

21.  logf^^*.    23.  log 7^2.  25.  log?^.      27.  log^. 

403.    To  prove  the  relation 

logj  m  = 


Assume  the  equations 

a'  =  m]        ,  ( x  =  loga  m, 

[ :  whence,  \ 
b"  =  m  )  [y  =  logt  m. 

From  the  assumed  equations,  a'  =  b^. 

X 

Taking  the  yth  root  of  both  members,  W  =  6. 
Therefore,  log„  h  =-,  ov  y  = 

y 

That  is,  logj  m 


y  log„  h 

loga  m 


log.  6 


LOGARITHMS.  345 

404.  To  prove  the  relation 

Putting  m  =  a  in  the  result  of  §  403,  we  have 

Whence,  log^  a  x  log^  h  =  l. 

405.  In  the  co^nmon  system,  the  mantisscB  of  the  logarithms 
of  numbers  having  the  same  sequence  of  figures  are  equal. 

Suppose,  for  example,  that  log  3.053  =  .4847. 

Then,  log  305.3  =  log  (100  x  3.053)=  log  100  +  log  3.053 

=  2  +  .4847  =  2.4847 ; 
log  .03053  =  log  (.01  X  3.053)=  log  .01  4-  log  3.053 

=  8  -  10  +  .4847  =  8.4847  -  10 ;  etc. 

It  is  evident  from  the  above  that,  if  a  number  be  multi- 
plied or  divided  by  any  integral  power  of  10,  producing 
another  number  with  the  same  sequence  of  figures,  the 
mantissae  of  their  logarithms  will  be  equal. 

The  reason  will  now  be  seen  for  the  statement  made  in 
§  390,  that  only  the  mantissas  are  given  in  a  table  of  logar 
rithms  of  numbers. 

For,  to  find  the  logarithm  of  any  number,  we  have  only 
to  take  frojn  the  table  the  mantissa  corresponding  to  its 
sequence  of  figures,  and  the  characteristic  may  then  be  pre- 
fixed in  accordance  with  the  rules  of  §§  391  and  392. 

Thus,  if  log  3.053  =  .4847,  then 

log  30.53  =  1.4847,  log  .3053     =  9.4847  -  10, 

log  305.3  =  2.4847,  log  .03053    =  8.4847  -  10, 

log  3053.  =  3.4847,  log  .003053  =  7.4847  -  10,  etc. 

This  property  is  only  enjoyed  by  the  common  system  of 
logarithms,  and  constitutes  its  superiority  over  others  for 
the  purposes  of  numerical  computation. 


346  ALGEBRA. 

406.  1.  Given  log  2  =  .3010,  log  3  =  .4771 ;  find  log  .00432. 

We  have,  log  432  =  log  (2^  x  S^)  =  4  log  2  +  3  log  3  =  2.6353. 
Then  hy  §  405,  the  mantissa  of  the  result  is  .6353. 
Whence  by  §  392,  log  .00432  =  7.6353  -  10,  Ans. 

EXAMPLES. 
Given  log  2  =  .3010,  log  3  =  .4771,  and  log  7  =  .8451,  find : 

2.  log  2.8.  7.   log  .00375.   '  12.  log  2.592. 

3.  log  11.2.  8.   log  6750.     ■  13.  log  274.4. 

4.  log  .63.  9.  log  .0392.  14.  log  (3.5/. 
6.  log  .098.  10.  log  .000343.  15.  log  a/61. 
6.   log  32.4.            11.   log  .875.   •  16.  log  (12.6)1 

USE  OF  THE  TABLE. 

407.  The  table  (pages  348  and  349)  gives  the  mantissse 
of  the  logarithms  of  all  integers  from  100  to  1000,  calculated 
to  four  places  of  decimals. 

408.  To  find  the  logarithm  of  a  number  of  three  figures. 
Look  in  the  column  headed  "  No."  for  the  first  two  sig- 
nificant figures  of  the  given  number. 

Then  the  mantissa  required  will  be  found  in  the  corre- 
sponding horizontal  line,  in  the  vertical  column  headed  by 
the  third  figure  of  the  number. 

Finally,  prefix  the  characteristic  in  accordance  with  the 
rules  of  §§  391  or  392. 

For  example,         log  168  =  2.2253 ; 

log  .344  =  9.5366 -10;  etc. 

409.  For  a  number  consisting  of  one  or  two  significant 
figures,  the  column  headed  0  may  be  used. 

Thus,  let  it  be  required  to  find  log  83  and  log  9. 


LOGARITHMS.  347 

By  §  405,  log  83  has  the  same  mantissa  as  log  830,  and 
log  9  the  same  mantissa  as  log  900. 

Hence,  log  83  =  1.9191,  and  log  9  =  0.9542. 

410.  To  find  the  logarithm  of  a  number  of  more  than  three 
figures. 

Let  it  be  required  to  find  the  logarithm  of  327.6. 
From  the  table,        log  327  =  2.5145, 
and  log  328  =  2.5159. 

That  is,  an  increase  of  one  unit  in  the  number  produces 
an  increase  of  .0014  in  the  logarithm. 

Therefore,  an  increase  of  .6  of  a  unit  in  the  number  will 
produce  an  increase  of  .6  x  .0014  in  the  logarithm,  ox  .0008 
to  the  nearest  fourth  decimal  place. 

Whence,      log  327.6  =  2.5145  +  .0008  =  2.5153. 

Note.  The  difference  between  any  mantissa  in  the  table  and  the 
mantissa  of  the  next  higher  number  of  three  figures  is  called  the  tab- 
ular difference.     The  subtraction  may  be  performed  mentally. 

The  following  rule  is  derived  from  the  above : 

Find  from  the  table  the  mantissa  of  the  first  three  significant 
figures,  and  the  tabular  difference. 

Multiply  the  latter  by  the  remaining  figures  of  the  number, 
with  a  decimal  point  before  them. 

Add  the  result  to  the  mantissa  of  the  first  three  figures,  and 
prefix  the  proper  characteristic. 

EXAMPLES. 

411.  1.   Find  log  .021508. 

Tabular  difference  =     21  Mantissa  of  215  =  33?4 

.08  2 

Correction    =  1.68  =  2,  nearly.  3326 

Result,  8.3326  -  10. 


348 


ALGEBRA. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

cx)86 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1 106 

13 

1 139 

^ns 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

I46I 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

I76I 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

26^5 
2^6 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

561 1 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

^Hl 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42  6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43  1  6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340  7348 

7356 

7364 

7372 

7380 

7388 

7396 

No. 

0 

1 

2   3 

4 

5 

6 

7 

8 

9 

LOGARITHMS. 


349 


No. 

0 

1 

2 

3   4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427  7435 

7443 

745J 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505  [7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582  7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973  7980 

7987 

63 

7993 

8000 

8cx)7 

8014 

8021 

8028 

8035 

8041  8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109  8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432  ^  8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8693 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8669 

8675 

8681 

8686 

74 

8692 

869^ 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9233 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956  1  9961 

1 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 
9 

No. 

0  i  1 

2 

3  ,  4 

5 

6 

7 

8 

350  ALGEBRA. 

Find  the  logarithms  of  the  following: 

2.  53.  6.   1068.  10.   7.803.  14.  4072.6. 

3.  2.6.  7.   82.95.         11.    .0003787.  15.  .0064685. 

4.  871.  8.    .9616.         12.   253.07.  16.  .013592. 

5.  .689.  9.   .007254.      13.   .91873.  17.  4.0354. 

412.    To  find  the  iiumher  corresponding  to  a  logarithm. 

1.  Eequired  the  number  whose  logarithm  is  1.6571. 
Find  in  the  table  the  mantissa  6571. 

In  the  corresponding  line,  in  the  column  headed  "  No.," 
we  find  45,  the  first  two  figures  of  the  required  number,  and 
at  the  head  of  the  column  we  find  4,  the  third  figure. 

Since  the  characteristic  is  1,  there  must  be  two  places  to 
the  left  of  the  decimal  point  (§  391). 

Hence,  the  number  corresponding  to  1.6571  is  45.4. 

2.  Eequired  the  number  whose  logarithm  is  2.3934.     ' 
We  find  in  the  table  the  mantissse  3927  and  3945,  whose 

corresponding  numbers  are  247  and  248,  respectively. 

That  is,  an  increase  of  18  in  the  mantissa  produces  an 
increase  of  one  unit  in  the  number  corresponding. 

Therefore,  an  increase  of  7  in  the  mantissa  will  produce 
an  increase  of  -J-^  of  a  unit  in  the  number,  or  .39,  nearly. 

Hence,  the  number  corresponding  is  247  +  .39,  or  247.39. 

The  following  rule  is  derived  from  the  above : 

Find  from  the  table  the  next  less  maMissa,  the  three  figures 
corresponding,  and  the  tabular  difference. 

Subtract  the  next  less  from  the  given  mantissa,  and  divide 
the  remainder  by  the  tabular  difference. 

Annex  the  quotient  to  the  first  three  figures  of  the  number, 
arid  point  off  the  result. 

Note.  The  rules  for  pointing  off  are  the  reverse  of  the  rules  for 
characteristic  given  in  §§  391  and  392. 


LOGARITHMS.  351 

I.  If  -  10  is  not  written  after  the  maiitissa,  add  1  to  the  character- 
istic^ giving  the  nmnber  of  places  to  the  left  of  the  decimal  point. 

II.  If  —  \0  is  written  after  the  mantissa,  subtract  the  positive  part 
of  the  characteristic  from  9,  giving  the  number  of  ciphers  to  be  placed 
between  the  decimal  point  and  first  significant  figure. 

EXAMPLES. 

413.  1.    Find  the  number  whose  logarithm  is  8.5264  —  10. 

5264 
Next  less  mantissa  =  5263  ;  three  figures  corresponding,  336. 

Tabular  difference,  13)1.000(.077  =  .08,  nearly. 
91_ 
90 
According  to  the  rule  of  §  412,  there  will  be  one  cipher  between  the 
decimal  point  and  first  significant  figure. 

Hence,  the  number  corresponding  =  .033608,  Ans. 

Find  the  numbers  corresponding  to  the  following  loga- 
rithms : 

2."  0.3075.  7.  9.9108  -  10.  12.  7.5862  -  10. 

3.  8.7284-10.  8.  7.6899-10.  13.  9.7043-10. 

4.  1.8079.  9.  0.8744.  14.  2.5524. 

5.  3.3565.  10.  8.9645-10.  15.  4.2306. 

6.  2.6639.  11.  1.8077.  16.  6.2998-10. 

APPLICATIONS. 

414.  The  approximate  value  of  an  arithmetical  quantity, 
in  which  the  operations  indicated  involve  only  multiplica- 
tion, division,  involution,  or  evolution,  may  be  conveniently 
found  by  logarithms. 

The  utility  of  the  process  consists  in  the  fact  that  addi- 
tion takes  the  place  of  multiplication,  subtraction  of  division, 
multiplication  of  involution,  and  division  of  evolution. 

Note.  In  computations  with  four-place  logarithms,  the  results 
cannot  usually  be  depended  upon  to  more  than  four  significant  fig- 
ures. 


352  ALGEBRA. 

415.   1.    Find  the  value  of  .0631  x  7.208  x  .51272. 

By  §  396, 

log  (.0631  X  7.208  X  .51272)  =  log  .0631  +  log 7.208  +  log. 51272. 

log   .0631  =    8.8000  -  10 

log   7.208=    0.8578 

log  .51272=    9.7099-10 

Adding,  log  of  result  =  19.3677  -  20  =  9.3677  -  10.  (See  Note  1.) 

Number  corresponding  to  9.3677  -  10  =  .2332,  Ans. 

Note  1.  If  the  sum  is  a  negative  logarithm,  it  should  be  written 
in  such  a  form  that  the  negative  portion  of  the  characteristic  may 
be  -10. 

Thus,  19.3677  -  20  is  written  in  the  form  9.3677  -  10. 

2.  Find  the  value  of  ?^. 

By  §  398,  log  ^  =  log  336.8  -  log  7984. 

log 336. 8  =  12. 5273  -  10  (See  Note  2.) 

log  7984  =    3.9022 

Subtracting,       log  of  result  =    8.6251  -  10 

Number  corresponding  =  .04218,  Ans. 

Note  2.  To  subtract  a  greater  logarithm  from  a  less,  or  to  sub- 
tract a  negative  logarithm  from  a  positive,  increase  the  characteristic 
of  the  minuend  by  10,  writing  —  10  after  the  mantissa  to  compensate. 

Thus,  to  subtract  3.9022  from  2.5273,  write  the  minuend  in  the  form 
12.5273  -  10  ;  subtracting  3.9022  from  this,  the  result  is  8.6251  -  10. 

3.  Find  the  value  of  (.07396)^ 

By  §  400,  log  (.07396)5  =  5  x  log  .07396. 

log  .07396  =  8.8690 -10 
5 


44.3450  -  50  =  4.3450  -  10  (See  Note  1 .) 

Number  corresponding  =  .000002213,  Ans. 


LOGARITHMS.  353 


4.    Find  the  value  of  V.0ao063. 

By  §  401,  log  v^.035063  =:  1  log  .035063. 

Iog.035063  =  a5449-10 

3)28!54'49  -  SQ  (See  Note  3.) 

9.5150  -  16 
Number  corresponding  =  .3274,  Ans. 

Note  3.  To  divide  a  negative  logarithm,  write  it  in  such  a  form 
that  the  negative  portion  of  the  characteristic  may  be  exactly  divisible 
by  the  divisor,  with  —  10  as  the  quotient. 

Thus,  to  divide  8.5449  —  10  by  3,  we  write  the  logarithm  in  the 
form  28.5449  -  30.     Dividing  this  by  3,  the  quotient  is  9.5150  -  10. 


ARITHMETICAL  COMPLEMENT. 

416.  The  Arithmetical  Complement  of  the  logarithm  of  a 
number,  or,  briefly,  the  Cologaritlim  of  the  number,  is  the 
logarithm  of  the  reciprocal  of  that  number. 

Thus,         colog  409  =  log  -i-  =  log  1  -  log  409. 

409 

log  1  =  10  - 10  (Note  2,  §  415.) 

log  409=    2.6117 

.-.  colog  409=    7.3883-10. 

Again,       colog  .067  =  log  ——  =  log  1  —  log  .067. 

.067 

logl  =  10  -10 

log  .067=    8.8261-10 

.-.  colog  .067=    1.1739. 

It  follows  from  the  above  that  the  cologarithm  of  a  num- 
ber may  be  found  by  subtracting  its  logarithm  from  10  —  10. 

Note.  The  cologarithm  may  be  obtained  by  subtracting  the  last 
significant  figure  of  the  logarithm  from  10  and  each  of  the  others  from 
9,  —  10  being  written  after  the  result  in  the  case  of  a  positive  loga- 
rithm. 


354  ALGEBRA. 

.51384 


417.   Examx)le.     Find  the  value  of 


8.709  X  .0946 


log  _^13§L_  ^  log  f. 51384  X -1- X -^\ 
^  8.709  X. 0946        ^\  8.709      .094(5; 


=  log  .51384  +  log  ^—  +  log     ^ 


8.709        "  .0946 
=  log. 51384  +  colog  8.709  +  colog.0946. 

.log.51384:=  9.7109 -10 
colog8.709  =  9.0601  -  10 
colog  .0946  =  1.0241 

9.7951  -  10  =  log  .6239,  Ans. 

It  is  evident  from  the  above  example  that  the  logarithm 
of  a  fraction  is  equal  to  the  logarithm  of  the  numerator  plus 
the  cologarithm  of  the  denominator. 

Or  in  general,  to  find  the  logarithm  of  a  fraction  whose 
terms  are  composed  of  factors, 

Add  together  the  logarithms  of  the  factors  of  the  numerator, 
and  the  cologarithms  of  the  factors  of  the  denominator. 

Note.  The  vahie  of  the  above  fraction  may  be  found  without 
using  cologarithms,  by  the  following  formula : 

log ^1^§^ =  log .  51384  -  log  (8. 709  x  .0946) 

^  8. 709  X. 0946        ^  ^^  ^ 

=  log. 51384  -  (log 8.709  +  log .0946). 

The  advantage  in  the  use  of  cologarithms  is  that  the  written  work 
of  computation  is  exhibited  in  a  more  compact  form. 


EXAMPLES. 

Note.  A  negative  quantity  has  no  common  logarithm  (§  387, 
Note).  If  such  quantities  occur  in  computation,  they  should  be 
treated  as  if  they  were  positive,  and  the  sig?i  of  the  result  determined 
irrespective  of  the  logarithmic  work. 

Thus,  in  Ex,  3,  §  418,  the  value  of  847.5  x  (-  2.2807)  is  obtained 
by  finding  the  value  of  847.5  x  2.2807,  and  putting  a  negative  sign 
before  the  result.     See  also  Ex.  34. 


LOGARITHMS.  355 

418.   Find  by  logarithms  the  values  of  the  following: 

1.   3.142x60.39.  4.    (- 4.3918)  x(- .070376). 

2..  541.21  X  .01523.  5.   .93653  x  .0031785. 

3.   847.5  x  (-  2.2807).         6.    (-  .00017435)  x  69.571. 

-    486.7  g        .5394  ..     9563.2 

76.51'  *    -.09216*  *    42712* 

g    1.0547  ,Q       2.708  -„    -  -00006802 

34.946*  *    .0086819*  '       .0071264 

^3    3.8961  x  .6945  ^^  (- .87028)  x  3.74 

4694  X  .00457  '  '    (- .06589)  x(- 42.318)* 

.-      718  x(- .02415)  .g    .09213  x(- 73.36) 

(- .5157)  X  1420.6*  '        .832x2808.7 

17.  (7.795)^  22.  (.7)1  27.  V^9. 

18.  (.8328)^.  23.  (-964)^.  28.  ^100. 

19.  (-25.14)3.  24.  (.00105)^  29.  a/1994. 

20.  (.03512)1  25.  V5.  30.  ^:07256. 

21.  lOl  26.  ^/2.  31.  ^.002613. 

32.    a/ -.00095173. 

33.   Find  the  value  of  ?^. 
3^ 

log  ^  =  log  2  +  log  v/5  +  cologS^  (§  417) 

=  log  2  +  1  log  5  +  ^  colog  3. 
o  6 

log  2=    .3010 

log  5=    .6990;  divide  by  3  =    .2330 

colog  3'=  9.5229  -  10  ;  multiply  by  -  =  9.6024  -  10 

6      

.1364  =  log  1.369,  Ans. 


356  ALGEBRA. 

34.    Find  the  value  of  ^ 


-  .03296 
7.962 


log  ;/i0§296  ^  1  1      ,03296  ^  1  ^^329^  _  ^      ^.ggg). 

^   >  7.962       3     "^  7.962       3  ^    '^ 


log  .03296  =  8.5180  -  10 
log    7.962  =  0.9010 


3)27.6170  -  30 

9.2057  -  10  =  log  .1606. 

Result,  -/1606. 


Find  the  values  of  the  following : 

.4 


36.   4^x7l 

3t 


10/79 
46* 


38. 
39. 
40. 


</7 


6/3        s/' 


36. 

37. 

44. 
45. 
46. 

47.    ., 

31  X  .414 

43     ^.0009657 


'.08 


(-10)^ 

/     4400\^ 
V     6937;  * 


41. 
42. 
43. 


/276.8 
\  940  " 

-</1000 

(-.6)^ 


V3x</5x  Vl. 
76  X  .0592\^ 


/76> 
V     1 

4 


307     J 


7.543 


50.  (25.467)i«  X  (-  .062)12. 

51.  a/5106.5  X  .00003109. 

52.  (83.74  X  .009433)7. 

53.  (4.8671)^  X  (.17543)i 
g^  A./3:928  X  A/eKi^ 

55. 


V72L32 

(.573)^ 


49. 


V.004978 
-  (.25691) 


8693.8  X  </.03307 


56. 


(-.0001916)^  xV68.1 
-  .27556 


.83457)^ 

57.    V^374  X  v':05286  x  \/.0078359. 
38.014 


58. 


-v/.04142x(-.947^) 


LOGARlTliJMS.  357 

EXPONENTIAL   EQUATIONS. 
419.   An  Exponential  Equation  is  an  equation  of  the  form 

To  solve  an  equation  of  this  form,  take  the  logarithms  of 
both  members ;  the  result  will  be  an  equation  which  ean  be 
solved  by  ordinary  algebraic  methods. 

1.  Given  31*  =  23 ;  find  the  value  of  x. 
Taking  the  logarithms  of  both  members, 

log  (31^)  =  log  23. 
Whence,  x  log  31  =  log  23  (§  400) . 

Therefore,  ^  =  !^  =  HItI  =  -^^^OS,  Ans. 

log  31      1.4914 

2.  Given  .2*  =  3 ;  find  the  value  of  x. 

Taking  the  logarithms  of  both  members, 
X  log  .2  =  log  3, 

Whence,         x  =  Mi  =       'f'^^       =  ^^  =  -  .6825,  Ans. 
log. 2     9.3010-10      -.6990 

EXAMPLES. 
Solve  the  following  equations : 

3.  332- =  5.17.      5.   .0158' =  .008295.      7.   a' =  6V. 

4.  .416^  =  6.72.      6.   5.336*  =  .744.  8.   mV  =  n. 

9.   7^-3  =.02041.  10.   .8''-^  =  .4096. 

11.  Given  a,  r,  and  Z;  derive  the  formula  for  n  (§  346). 

12.  Given  a,  r,  and  S ;  derive  the  formula  for  n. 

13.  Given  a,  Z,  and  S ;  derive  the  formula  for  n. 

14.  Given  ?',  I,  and  S ;  derive  the  formula  for  n. 

420.   1.   Find  the  logarithm  of  .3  to  the  base  7. 

By  §  403,  log,  .3  =  !^1^  =  MLD^^  ^  -  -5229  ^  _  gjg      ^„, 
•"*  *  log,„7  .8451 


35^  ALGEBRA. 


<)Ki 


X  'N  "^ /7}  EXAMPLES. 

Find  the  values  of  the  following : 

2.  log2l3.  4.   log.68  2.9.  6.   logi.6.838. 

3.  logs  .9.  5.   log.3,.076.  7.   log83  5.2. 

Examples  like  the  above  may  be  solved  by  inspection  if 
the  number  can  be  expressed  as  an  exact  power  of  the  base. 

8.  Find  the  logarithm  of  128  to  the  base  16. 
Let  logie  128  =  x  ;  then  by  §  387,  16^  =  128. 
That  is,  (24)^  =  2^,  or  2**  =  2\ 

7 
Whence  by  inspection,  4aj  =  7,  ov  x  =  — 

4 

Therefore,  logie  128  =  -,  Ans. 
4 

9.  Find  the  logarithm  of  81  to  the  base  3. 

10.  Find  the  logarithm  of  32  to  the  base  8. 

11.  Find  the  logarithm  of  ^  to  the  base  27. 

12.  Find  the  logarithm  of  -gij  to  the  base  ^. 


ANSWEES. 


§  5;  pagres  4  to  6. 

4.  56,  14.  6.    A,  49 ;  B,  07.  6.    95,  28. 

7.  A,  64;  B,  38.  8.    A,  $35;  B,  ^68. 

9.  A,$48;B,  $8.  10.    52,33.  11.    A,  $18 ;  B,  $.54 

12.  Men,  300;  women,  100;  children,  25.  13.    $0.99. 

14.  A,  $48;  B,  $32;  C,  $16.  15.    52,29,87. 

16.  A,  $78  ;  B,  $57  ;  C,  $95.  17.    Watch,  $48  ;  chain,  $8. 

18.  13,  26,  130.  19.    $54,  $18,  $72. 

20.  Cow,  $55;  sheep,  $18;  hog,  $11.  21.   36,  19,  72. 

22.  Ai  322  ;  B,  186.  23.    A,  33 ;  B,  50  ;  C,  18. 

24.  A,  $96;  B,  $24;  C,  $54. 

25.  Horse,  $240;  carriage,  $192;  harness,  $24. 
26.  $25,  $5,  $125.  27.  A,  140;  B,  168;  C,  204. 
28.  A,  $12;  B,  $7;  C,  $19;  D,  $31.  29.    4,  12,  36,  108. 

§  8;  pages  7,  8. 


2.   8. 

6. 

19 
30* 

10.   0. 

15.   64. 

19.   0. 

-f 

3.  360. 

7. 

27 
25' 

"•i- 

16.   324. 

20.   8. 

24.    1. 

4.   46. 

8. 

5184 

•    -f- 

17.   642. 

-   I 

25.   16. 

••f- 

9. 

73 
30' 

-  f  • 

§  16; 

18.    284. 
page  12. 

22    1^. 
6 

oc    39 
26.   -. 

"•i^- 

12.    - 

-f  •      -  «^ 

14. 

-6irV 

16.  7j 

§  19; 

page  14. 

-i- 

-  -^- 

14. 
1 

- 

24. 

15.   f. 

2  ALGEBRA. 

§  34;  pages  20,  21. 
2.  a -46.  3.    -2m2  +  3n3.  4.    -Gab-Ucd. 

5.  9a-4&-6c.       6.  4?7i2.       7.  ^^  _  ,/ _  ^.      S.  ^a"^ -Sab  -  2b'\ 
9.Sx^-x-4.       10.  5a  +  46-2c.       11.  a:^  -  8  a;^?/ -  2  a;i/2  -  3  j/s. 

12.  0.  13.   3a3-5a2  +  4a-2.  14.    6a2  -  3  62  _  5f^2. 
15.    a^  -  x^                       16.    7  a;3  +  22  x2  -  14  X  -  24. 

§  39;  page  23. 
25.    -37  a.  26.    bxy.  27.    -  14a2.  28.   34w3a:. 

§  40;  pages  24,  25. 

2.  4a2^3rt-20.  3.   3a6-66c  +  ca.  4.    -4a;y. 

5.  -66  + 8c.  6.    x^-x^-6x-^".  7.    -3ic  +  3y-3^. 

8.  6a -12  6  +  21c  +  2d  9.    -  9a3  +  8a2  -  4a  +  3. 

10.  8a;3+ 11x2-5.       11.  4a -2a2 -2  a^.       12.  5a2  +  6a6  -  5662. 

13.  10x3-6x2  +  9x-12.  14.  Ga^  +  3a26  -  12a62  -  8  63. 

15.  2x3  -  5x2?/  +  x?/2.  16.    7  a  -  6  -  8  c  -  4d. 

17.  5 -4x  + 7x2 -20x3 -5x4.  ig.  a:2  -  4x?/ -  6^2  +  7^^  ^  21y. 

19.  4  a^  +  10  a*  -  11  a3  -  16  a2  -  8  a  +  1. 

20.  x^-5x*y  +  6  x^y^  +  11  x^-y^  -  15  xy^  +  y^. 

21.  4a2.  22.  2a2-a6.  23.5x^-8x2-9.  24.  7x-6?/. 

25.0.  26.  3a  +  36  +  3c  +  3(Z.  27.  12a3  -  8a  -  7. 

§  43;   pages  27,  28. 

3.  5  a  +  12  6.         4.  7  m  -  3  «.  5.  x  +  y-3z.         6.  Sa'^-ah. 
7.  -  2  m2  +  n2.     8.  2  x  -  1.     9.  a-b-hc  +  d-e.      10.  -  2  a6  +  3. 

11.  8x-7.  12.  0.  13.    -10.  14.    -4.  15.    -lOx+1. 

16.  x-^y  +  z.        17.-3/1-5.         18.  17.        19.  3  a  -  1.        20.  0. 

21.   -2x-\-y-2z.  22.  x-y.  23.  1. 

§  52 ;  pages  33  to  35. 
3.  6a2  +  29a  +  35.     4.  30a2-53a  +  8.      6.  -32x2- 52 x?/- 15  2^2. 

6.  28  a262  +  34  a6  -  12.        7.  x^  +  y\        8.  10  a^  +  33  a2  -  52  a  +  9. 

9.  12 x3- 13x2+ 19 X- 12.      10.  5n3+2n2-19n-6.      11.  27a3-863. 

12.  a2  -  2  a6  +  2  ac  +  62  -  2  6c  +  c2. 

13.  12w5  + 8m%-31m3w2-24»»2^3. 


ANSWERS.  3 

14.  3x*+5x3-33x2  +  10x  +  24.  15.  m*  +  m^n^ -f  n*. 

16.  16  a*  -  1.  17.  63  X*  +  114  a;"*  +  49x2  -  16x  -  20. 

18.  8  n*  _  50  n2  ^  32.        19.  12  a»  -  47  a^b  -  8  a^b^  +  107  ab^  +  56  b*. 
20.  2x2  -  8^2  ^  24yz  -  ISz^.  21.  8 a2  +  40 ac  -  18  62  +  soc*. 

22.  a5  _  6  052  _  flj  _  G.  23.  x^  -  32.  24.  m^n  -  mn^. 

25.  10x5- 13x*- 52x3  +  26x2  + 58x- 9. 

26.  8X*"*  V*+^  -  22x2'»+2?/3n+l  -j.  15x6y5»-3 

27.  6  m5  -  13  m^  +  4  m^  -f  9  «i2  _  1 1  wi  +  3.  28.  32  a^  +  243. 

29.  ««  -  5  a*6  +  10  a^b'^  -  10  ah"^  +  5 a6*  -  t^.      30.  x^  -  6  x*  -  3 x2  -  1. 
31.  a6-12a*  +  48a2-64.  32.  w&-8«i<tt  +  48m-u3  +  ii  „i;i4_28n6. 

33.  x«  -  6x*  +  13x2  -  9.  34.  a^  -Zabc-b'^-  c^. 

'     35.    12  x5  -  2  %^y  -  22  x^y^  +  9  x^y^  +  8  xy*  -  4  y^. 
36.  x3  -  9x2  +  26x  -  24.  37.  ^a^  +  26a2  -  67  a  +  15. 

38.  x6  -  j/«.  39.  60  n^  -  127  11^  -  214  w  +  336.  40.  a^  -  x\ 

41.  4  m*  -  73  w»2n2  +  144  n*.  42.  a»  -  1.  43.  x8  +  x*+l. 

44.  4a*-13a262+9  6*.  45.  16x6-144x4-x2+9. 

§  53;   pages  35,  36. 

2.  11  x2  -  1 11.  3.  2  a.  4.  2  ab  -  2  win.  6.    -  4  xy  +  4  xz. 

6.  a2  _|.  52  _|.  c2  4.  ffJ  _  2  a&  -  2  ac  +  2  ad  +  2  6c  -  2  6d  -  2  c<Z, 

7.  16 x*  -  72 x2  +  81.  8.  2  a%  -  2  ab\  9.  4 x2.  10.  0^  +  2  a^x3+  x^. 
11.  a8  -  68.  12.  12  x2  +  12.  13.  -  x2  -  y2  _  ^i  4.  xy  +  y«  +  zx. 
14.  0.         15.  16a3-2a.         16.  3x- +  3i/2  +  3^2  _  2x?/ -  2y2r  -  2  2X. 

17.  4  a*  -  64  x4.  18.  8  be.          19.  6  m^  +  16  ?7i^»  +  16  imv^  -  6  ji*. 

20.  -  a8  -  63  _  c3  +  a25  _|.  ^52  _|.  052^  +  ctc2  +  62c  +  6c2  -  2  a6c. 

21.  6a2ft4.2  63.  22.   -2x^ -2y^  -  2z^  +  6xyz. 

§  61 ;   pages  42  to  44. 

3.  5x  -  7.  4.  5 m  +  4  n.  5.  2  a  -  3.  6.  x2  4.  a;  _  12. 
7.  4 m2  -  6  win  +  9  )A               8.  x2  +  4xy  +  16  y\  9.  2  a  -  4. 

10.   -lO.ry-6.       11.  5a26+6a62.       12.  m^-mn-Sn^.      13.  3«  +  4. 
14.  2a26-a62.      15.  a-6  +  c.      16.  2x-4y.       17.  5?n2-3mM+4  n2. 
v   18.  2a2-3a  +  5.     19.  x2  +  2x+l.    20.  w-2.    -21.  2m^-^>m''--3Mj=l^ 

22.  x2+x?/+!/2.  23.  l-2a2+4a*-8rt6.  24.  8xH12x2f/  +  18xy2+27y3. 
25.  w2_3 771-4.      26.  3x2-x-2.      27.  a2+a-l.      28.  2x2+9x-5. 


4  ALGEBRA. 

29.  4m2-2w?i2+w*.     30.  x*-2x^+ix^-Sx+16.     31.  10 a^+Sa-i. 

32.  m^-1.  33.  a +  3.  34.  4  x'«+V  -  4  a: V- 

35.  2  a'^  +  2  a^b  +  2  a%^  +  2  «&«.  36.  a^  +  a26  +  ab^  +  ft^. 

37.  2m2-3.  38.  4a2-12a  +  9.  39.  2x3  +  5a:2  -  8a;  -  7. 
40.  x3  -  3x2  -  3.              41.  a2  _  2  a  +  10.  42.  x^  -6xy  +  9y^. 

43.  3x3  -  x2  -  2  X  -  5.  44.  2  a^  -  5 a2  _  6  a  +  4. 

45.  m^  —  2 m2«  -  mn^  +  2  n^.  46.  4a  +  b  -  c.  47.  x?  +  2/?  -  ^'•. 
49.  x-c.  50.  x2+(a  +  &)x  +  a&.  51.  x-2  6.  52.  (a  +  6)x-c. 
53.  (w  — n)x— j9.  54.  x  +  a.  55.  x2  —  (6  +  c)x  +  5c. 
66.  a(6-c)+d                   57.  a+(2m-3yi). 

§  62  ;   pages  45  to  47. 

2.  270.  3.    -9.  4.  42.  5.  729.         6.    -5.         7.    -— • 

15 

8.  -748.        9.  854.        10.  — •         11.   - -•         16.  9(x  +  t/)2-25. 
o  2 

17.  63(a  -  &)2  -  20(a  -b)-  32.  18.  2(m  +  n)+S. 

19.  (X  -  y)2  -(X  -  y)+  1.  21.  -V-a  -  ^jft  +  i§c. 

22.  -l^x  +  ify-T'iT^r.  23.  -la-j\b-{-ii-c. 

24-  -  tV*  ~  I y  -  H^'  25.  ^7X3  -  -27. 

26.  ija^--j\a^b-hj\ab^-^\b^.  27.  fx2-|x  +  .V 

28.    I  ^2  _  I  ^6  +  ^  62.  29.    G54p54  _  2  a2p+3533+2  +  Qj656g. 

30.  x'«+i  -  x3?/2«+i.   31.  a2i'+3  +  ai'+2623-i  +  a6<?-2.     32.  2(x  +  1)2  -  3. 

33.  -5(x+y)2-10x(x+y)  +  15.         34.  8x-2.         35.  fx2-|x-i^. 

36.  x3  +  (a  +  6  -  c)x2  +  (a5  -be-  ca)x  -  abc.      37.  a^+%^  +  a2&'»-i. 

38.  -ia2+MQ5_j3.  39.    (wi_,i)4_2(,n_n)2+ 1. 
40.    a3«+i62  +  a63«+2.                   41.    4  ^2.  42.   0. 

43.     ^9_^4_|aj3_   7^2+4^  +  .^.. 

44.  f  a2  -  I  a  +  \.  45.    a^  -  3  a26  +  3  a62  _  53. 

46.  3  xS'^-iyS  -  7  x2?/2'>+i.  47.  (a  +  6)x2  +  {a^  +  52^^^;  -2ab{a-\-  6). 
48.    (a  -  6)2  -  2  c(«  -  6)  +  c^.  49.    x2'«  /x«y»*  +  2/2n. 

50.  -4  a*  -  I  a2x2  -^ax^-  yV  a:*-     / 

51.  x^  +  (-  a  +  6  —  c)x2  +(—  aj/—  be  +  cd)x  +  a6c. 

52.  X2i'  +  X2?  +  X2'-  -  2  XP+?  +  2'XP+''  —  2  X9+»". 

53.  |x2  -  1  X  +  f.     54.  x2  +  (a  -  6)x  -  ab.     55.  x^  +  ?/3  +  ^3  -  3x^/0. 
56.   2  a252  +  2  62^2  +  2  c2a2  -  a^  -  6*  -  c*. 


ANSWERS. 

§  75  :  pagres  51,  52. 

3. 

14. 

8. 

2. 

13. 

5 
ll' 

18.    -6. 

24. 

2 

7* 

29. 

32 
11* 

4. 

-7. 

9. 

5 

7* 

14. 

4 
5' 

19.    5. 

25. 

4. 

30. 

_1 
4' 

5. 

4. 

10. 

- 

4 

3* 

15. 

3. 

21.    -5. 

26. 

7 
5' 

31. 

-1, 

6. 

-5. 

11. 

1. 

16. 

8 
9' 

22.    2. 

27. 

1, 
2* 

82. 

10 
3* 

7. 

-9. 

12. 

2 
3" 

17. 

8. 

23.    -10. 

28. 

-6. 

§  77 ;  pagres  55  to  58. 

6.    10,9.     6.    169,87.     7.    24,14.    8.  A,  $7.50 ;  B,  $5.25 ;  C,  $9.25. 

9.    A,  65  ;  B,  13.  10.    A,  42  ;  B,  84.  11.    A,  $  12  ;  B,  $36. 

12.  9  five-cent  pieces,  7  twenty-five  cent  pieces.  13.  8.  14.  17. 
15.  6  fifty-cent  pieces,  11  dimes.  16.  47,  29.  17.  9,  4.  18.  13,  7. 
19.    A,  43 ;  B,  57.        .  20.    9  oxen,  27  cows. 

21.  3  dollars,  12  dimes,  15  cents. 

22.  3750  infantry,  500  cavalry,  125  artillery. 

23.    A,  320 ;  B,  1600  ;  C,  3840.    24.  A,  $25  ;  B,  $  18  ;  C,  $40 ;  D,  $32. 

25.  Wife,  $864  ;  each  son,  $72  ;  each  daughter,  $216. 

26.  A,  $42;  B,  $23;  C,  $29;  D,  $31. 

27.    13  three-penny  pieces,  36  farthings.    28.  44,  27.     29.  324  sq.  yd. 

30.    12.  31.    35,  36,  37.  32.    A,  68  ;  B,  18. 

33.   8  $  2  bills,  13  fifty-cent  pieces,  24  dimes.  34.    7,  8. 

35.   3,  4,  6,  6.  36.    Worked  22  days,  was  absent  10  days. 

37.  6  bushels  of  first  kind,  18  bushels  of  second  kind. 

38.  75  men  on  a  side  at  first ;  whole  number  of  men,  5668. 

39.  First  class,  75 ;  second,  115 ;  third,  150  ;  fourth,  195. 
40.    18.  41.   A,  8  minutes  ;  B,  5  minutes. 

42.    15  pounds  of  first  kind,  35  pounds  of  second  kind. 

§  82  ;  pagres  60,  61. 

25.  a2  -I-  2  ac  +  c'^  -  h\  30.    1  -  a^  -  2  a6  -  h'^. 

26.  x^-2xy-\-y^-z^.  31.   a;* -2x2  +  1. 

27.  rt2  -  62  _  2  6c  -  c2.  32.   a2  -  4  62  +  12  6c  -  9  c2. 

28.  a*-a2  +  2a-l.  33.    a4  +  a262+64. 

29.  x*-5a;'2  +  4.  34.    9^2  -  16y2  _  16?/^  -  4^2. 


6  ALGEBRA. 

§  99  ;  pag-es  72,  73. 

38.  (a-6  +  c)(a-6-c).  44.  (3  a-4  6  +  2c)(3a-4  &-2c), 

39.  (m-h  n+p){m  +  n -p).  45.  {4:X-\-2y-  5z)(4:X-2  y  +  5 z). 

40.  (a  +  x-\-y){a-x-  y).  46.  {m-2n-{-x)  (w  -  2  «  -  x).  • 

41.  (x  +  ?/-s)(x-?/  +  ;2;).  47.  (2a  +  6  +  3)(2a-6-3). 

42.  («  +  6  +  2)(a  +  6-2).  48.  {bx  -^  y  -\-Zz){hx  ^  y  -  ^z). 

43.  (1  +  w -«)(!- wi  +  n).  49.  (a-6  +  c-fZ)(a-6-c+d). 

50.  (a  +  X  +  6  -  ?/)(«  +  X  -  6+ ?/). 

51.  Cx  —  wi  +  y  +  n)  (x  —  m  —  2/  —  n). 

52.  {X  +  y  +  a  +  h){x  ^-  y  -  a  -  h). 

53.  (2  a  +  6  +  3  c  -  2)  (2  a  +  6  -  3  c  +  2). 
64.  (x  -  4  y  +  0  +  6)  (x  -  4  y  -  0  -  6). 

55.  (5  a  —  m  +  &  —  3  n)  (5  a  —  «i  —  6  +  3  n). 

§  106 ;  pages  78  to  80. 

30.  (1  +  702(1  -  ny.  45.  (2x  +  ^y)\2x-^yY. 

41.  (a  +  3)2(a  -  3)2.  46.   (a  -  l)2(a2  +  «  +  1)2. 

42.  (x+l)(x-2)(x2  +  x  +  2).       52.   (3  «  + 2)2(3  a  -  2)2. 

43.  (a  +  26X«-26Xc+3(?Xc-3fZ).   53.   (x  -  2)(x  +  3)(x -3)(x  +  4). 

44.  (x  +  l)(x-l)(x-4)(x-6).  54.   (a  -  1)*. 
55.  (a  -  x)(6  +  y)(a2  +  ax  +  x2)(62  _  &y  +  y'l), 

57.  (6a  +  2fo-7c)(6a-25  +  7c).  59.   (x  +  l)2(x  +  2)2. 

60.  (a  +  l)(a-2)(a2_a  +  i)(a24.2a  +  4). 

63.  2  &c(a  +  6  +  c)  (a  -  6  -  c). 

64.  (a-l)(rt  +  3)(a  +  4)(a  +  8).  66.   (x  -  l)(x  +  2)2(x  -  3). 
67.  {a-\-h-\-c){a-h  +  c)(a  +  6  -  c)(a  -  6  -  c). 

76.  (m  +  x)  (m2  -  4  «ix  +  7  x2).  77.  &(3  a2  -  3  a6  +  &2). 

78.  (x-y)(9x  +  y).  79.   (a  +  6)(a2  -  3a&  +  6'0- 

80.  (a  +  ?>  +  c  +  d)(a  +  &-c-(Z).  81.  2x(x2  +  3). 

82.  (x  + 2/) (2x2  +  2/2).  83.  (a  +  l)2(a-l)2(a2  +  i). 

84.  (a  +  X  +  6  -  ?/)(«  +  X  -  6  +  ?/).  85.  m(x  -  m)(m  -  3x). 

86.  2^/(3x2  +  2/2).  88.   (x  +  l)2(x-l)(x2+l)(x2-x+ 1). 

89.  3a(a-l).  90.  7(5m  -  l)(w2  -  »»  +  1). 

91.  {x  +  y -z){x-y  +  z)(x^y -\- z){x-y  -  z). 

92.  (a-5  6+4c+3d)(a-5&-4c-3cZ).  93.   (l  +  a)(3-a-a-). 


ANSWERS.  7 

§  117  ;  page  89. 
5.  a;  -  1.     6.  2  a  +  3.    7.  x  +  2.     8.  x  -  3.     9.  m  +  1.    10.  3  a  -  b, 
11.   3a-^  +  «x-2x2.  12.   a;(2x-5).  13.  3x  +  4y. 

14.   2  a^  -  3  a2  -  a  4-  4.  15.   2  )7i2  -  mn  +  n"-.  16.  x  -  2. 

17.  a'^  +  2a  +  4.  18.  m'2  -  2 mx  -  3x2.  19.  a  -  1.  20.  7n\m  +  2). 
21.  a  -  5  6.  22.  x  +  3.  23.  3  a^  -  2.  24.  a  +  4.  25.  2  x  -  y. 
26.    2x2-3x-l.  27.   x  -  2.  28.   ax{a  +  x). 

§  118  ;  page  90. 
^2.   2x-9.      3.   4a +  1.       4.    3m +  4.       5.   5a -26.       6.   x  +  2. 
7.   a  +  1.  8.   m-  1.  9.   2x-3y. 

§  125 ;  page  93. 

30.   (x  +  y  -h  z)(x  -  y  -\-  z)(x  -  y  -  z).  40.   (m  +  n)^  (m  -  n)2. 

41.  (^a  +  b  +  c)(a  -  b  -  c)(a  -{-  b  -  c). 

§  126 ;  pages  94,  95. 

2.  (2x  +  7)(2x2-19x  +  45).      5.  x^(ax  -  ?/)(8x2  +  21  xy  +  10y2). 

3.  (a-4)(3a2  +  14a-5).  6.  3(4m  +  5)(4  m^-ll  «i2-6?)i  +  9). 

4.  (3a+86)(12a2  +  16a6-362).  7.  (2a  +  3)(3a5  -  14a2  -  a  +  6). 

8.  x(2  a^  -  ax  +  3 x2)  (2  a^  +  5  a-x  +  2ax^-  x^). 

9.  (2 a  -  3 6)(a*  +  a^ft  -  5 a262  +  2 aft'  +  64). 

10.  (3x-2)(4x*-5x2+4x-3).  11.   (a2-3a  +  2)(4a3-9a-4). 

12.  2  wn(3  m2  -  mn  -  2  n^)  (3  m^  -  2  m^n  -  7  mn2  -  2  n^). 

13.  a2(a2  -  2  a  +  3) (3  a*  +  11  a3  -  6  a2  -  7  a  +  4) . 

14.  (x2-x-3)(3x4  +  7x3  +  6x2-2x-4). 

§  127  ;  page  95. 

1.  8x*  +  20x^-46x2-117x-45. 

2.  162  a*  +  117  a3  -  147  a^  -  62  a  +  40. 

3.  12  m*  -  10  to3  -  8G  ??i2  +  140  7rt  -  48. 

4.  24  x7  -  70  x6  -  15  x5  +  25  x*  +  G  x'. 

5.  a^  +  2  a*  -  10  a3  -  20  a2  +  9  a  +  18. 


12. 


§  133  ;  pages  98,  99. 

3a 
4  6* 

13-   ?1- 

5  2/2 

14    « +  2        j5    x(x-2) 
a-l                   x-6 

jg    5  a  +  2  6 
5a-26 

8  ALGEBRA. 


17        in  -S  j^g    X  +  y         jg    a(8ff  +  7  a;)  g^    a;  -  9?n 

m(w  — 0)  '     2xy  x(8a  — 7x)  '    x-\-Zm 

a-+2a  +  4       gg    2m-5      „,    x  +  y-^z      „^    U«--12«6  +  16 />2 


21 


25.    1.      26.  ^  +  ^  +  ^  +  ^.     27.   , 

a  —  b  +  c  —  d  Sx 

31.   ^^^. 

Z/  +  X 


5x+7'        '  2a-r        '  2ic-9y'        '  3??z+4       ""  x^-Sx+'l 

„     og-r  ^M  g     3x\2                    g     2q!  +  1 
'   3x  +  l'                  *   6a-l' 

,Q  ,jitr—_iii^rj^^  11      g^  +  3  qx  4-  x^ 

■   m2  +  4  m  -  2*  "    a2  -  2  ax  -  4x2* 

§  137  ;  page  102. 

5.  4x  — 6-^ 11.   3  a ■ — 

2x  +  3  4a-l 

6.  a;2  +  xy  +  2/2  +  ^i^.  12.   37n2  +  4-  '^  ^^  +  ^  . 

X  —  y  4  m^  +  1 

7.  a2-a6  +  &2_J^.  13,   a;^  -  x2y  +  x?/2  _  y3  +  Jj^. 

a-\-b  x  +  y 

8.  5rt2_3rt-l ? 14.    6a +  7           2a-3 


3. 

a- 

2a- 

-2 
-1* 

3  a  +  2  ?; 

4a 

-6 

7/l2 

-  m 

+  3 

3  a  +  4  *                     3  a-2  -  4  a  f  5 

9.   2m  +  5«  +  -^^-?^.  15.  a^-\ra^b  +  a'^b'^-^abHbH^' 
2  m  —  5  n                                                               a  —  o 

10.    2X-1+      ^'~^ —  16.  4x2  +  6x-2          ^^"^ 


x2  -  X  -  1  2  x2  +  X  -  3 

§  138  ;  page  103. 

g    3a2-lla  +  2       ^     2^       g    10a-^-13a-9       g    4x2-10x-7 
3a  x-y         '         2a -3  '  5x 

7    _?A_.     8    !^iilzJi!.     9       10^         10     ^^^'        11      ^-^'^ 


3a  +  &        *w  +  n  2a-5x          ■3x-4             x  +  2y 

j2     8m34-24m-^-36m-27  ^3    4a'^+5a        ^^    5a2-23ay>  +  86^ 

2TO  +  3               '  '     2a-l   '          ■          4a-3  6 

j^     2x2y  +  2ry2       jg    oM^t  ^^       -3x2         ^g             M  n^ 

x^+xy  +  ^Z"-^                «-&  x2+x+l           '  ?n--3?nn  +  9«- 


ANSWERS. 


6. 
10. 
12. 

8. 

10. 
13. 
19. 


Qx^-Sx 


§  140 ;  page  105. 


4a62_4  53 


2x(9x2-l)' 2x(9a;2-l)"       '     {a-b)(ia^-hb^)  \a-b){a^-{-b^) 
3(a-l)(a2  +  l)     Q(a  +  l)(a^  +  l)     9(a^  -  1) 
a*  -  1  '  a*-l         '      a*  -  1    * 


2  x3  -  16 


3a;S-f  6a;^  +  12a;S 


3x5_i2x* 


3x2(x-4)(x3-8)'   3x2(a;-4)(x3-8)'   3x2(x  -  4)(x3  -  8)" 
"      y2  (g  -  6)2 


11. 


(a-6)(x-y)2'   (a-6)(x-y)' 


(a+5)2 a2-9 q2_4 

(a+2)(a-3)(a  +  5)'  (a  +  2)(a-3)(a  +  5)'  (a+2)(a-3)(a  +  6) 


21a -4  . 

^    4x2  +  3m2 
96  m:*: 
20x8-4x2+57x+35 
40x3 


§  142 ;  pages  106  to  111. 
20x2- 18y  .     6x+l 


15  xV  48 

g    ab  ■\-bc+  ca 
abc 
4  6c-9ra+8a6 


g    3a2-14x2 

18a2x2 
10a  ^63 


28 


11. 


24rt6c 


12. 


3x-10w 


53  X 
36* 

m2-l 


14. 


20 


16. 


5a 


7X-22 


(2x+l)(5x-6) 


2m2  +  2n2 
m2  —  n2 
2x 

x-y 


24. 


108 
21. 

4x 
x2-l 

27 


18. 


30  X 
11a -9 


(3a  +  5)(4a-7) 

«!±^.      22    «V15a  +  3 
a2-62'  •    a[-2_3fl5_28" 

4a 


25. 

10  a?) 


4a2-l 


31. 


35.   - 


(x-2)(x  +  6)(x-9) 

32.  1. 

12a4-18 
a(a-3)(a+6) 


b 
a-\-b 

4x2 
(x+2)3' 


4x 

1  +  x* 


4^2 


^^^    .     40.  0.     41. 

8aH&^^  {x-yY 


42. 


(2a  +  36)2(2a-3  6) 

Q    10  ax 

(x-3a)2(x-7a)' 
2y2 

x2-?/2' 

8 
x+2" 
2n2 


33. 
37. 


(ni  —  n)^ 


44. 
48. 


45. 


a2-a4-l 


x-3 
49.  0. 


46. 


4a&2 


47. 


50. 


11 


(l+a;)(2-x)(3+x) 


•  a(a2-x2)* 

,Q    2x2?/  +  2xv2 

38-            ^S_y3 

13         7-3x 

(24-x)(4-x) 

a3  -  2  x3  ^ 

(a  +  x)(a3-x3) 

-       53.  -^^  +  ^V 

xy(x-y) 


10  ALGEBRA. 

54      lOx-1.  55    o:^.       56    4 5      ^               ^_ 

12(x-2)              a2-9               m(16-TO2)  1-a 

-        ^'  60.    ^^li^.               61.         ^  "-       2  m 


0:2  _  4                      ^  _j_  1/                       9  _  4  Qj2  ?)i  +  2 

64.  ^  +  ^f-       65.  ?^ 66.  0. 


§  144  ;   pages  112,  113. 

4.   -^.         5.  I         6.  2abc.         7.  ^.         8.  ^^. 
5m2  3  a3  32  y* 

9.    5(a4-6)  jQ   3m+l  ^^    2a:2(^-.3)  ^^    ^C^+^y) 

3(a  +  l)  ■    m-5  "      (a;-C)2  *    ac(ic+?/) 

13     (a-46)(a-2&)^       ^^    -^(^^  -  1)        .  15    l  le   -J^. 

rt(a-3&)                '  (x  +  2)(x2+a:  +  l)  '  2           '  a-l 

j^^   2x-3y^       jg^  (x  +  y-^)2^      ^g    («Z1^.  20    1.  21    ^±^. 
a;-y               (x-?/-2)2              (a  +  b)^                              2x 

§  146 ;   pages  114,  115. 

3       3g«              4    9m^^            5    3Ca;  +  3)             g  m(2w  +  5w) 

76xV'             '      4«*/              '2(x-2)'              ■  n(im-3n)' 

^    3(2a-56)            g    ff(a  +  7)           ^    a;Cx  +  2y)  ^^       a;2 

6(4a  +  36)'            ■    («-3)2*            '           ^         "  "  x2,~  1* 

jj     a(a-2)_       j2    (a  +  26)(a-56)         jg    2«  +  a;  ^^    ff  +  6  +  c 

a  +  5              '  (a  +  86)(rt  +  46)            '  a  +  2x  *  a-6  +  c 

§  148  ;  pages  115  to  117. 
J,         2  5    3^±x±l^  g    w-M  Y    2x-3" 


2  m  —  1  X  wi  —  H 

9.  ^+Ay.  10. «.       11.  ^  +  2^.       12. 

X  +  2  y  6  ^/ 

13.   ^Zll.  14.  ^'-^^'.  15.  a  +  1. 

X  +  y  a2  -  62 

jg     103x4-78  jg    2-3  a  g^    «  +  36 

'     39X+30*  ■    5-7a'  '    3a-6' 

22        ^     ■  23    2(^  +  y>.  24    ItiUL^lJll. 

'    1  +  x2  '    (x-yy  m                        a2  +  &2 


6 

^-         8.  X. 

(X- 

3)Cx  +  2) 

X 

16. 

2  a2  _  3  62 

7a6 

21 

2(x-«) 

X  4-  « 

S 

5.        «^ 

§  149 ;  pages  117  to  119. 

«262  ^  9X2 

4a+3  "■  a'^-ab  +  b^'         '  (2a-3x)2        "  i  +  ar  +  y  +  xy 


J     79(1-31        2  a'^ft^  3  9^^  A  1 


ANSWERS.  11 

1,        6.  2^-=^.        7.  «^^1^.        8.  0.        9. 


a2  +  1                  2  ic</                    a363  x  +  2  y 

,^    (a:-2)(a.-8).        ^^    q.        12.  4a^-9.        13.  ^a;^!^ 

bia^-ab-{-b'^)           '    a:V            "  ac-bd  '  (x-8)(3a;-8) 

18.    1.         19.  ^n(^'>^^'+n').        20.  ^-±-^.  21.  ^— •        22.  2 

w'^C?)*''^  —  71^)                 a  —  b  I  +  X- 

23.    («-^^^  24.  ^^'-^^-^^  25.   2 26.   1, 

2{a  +  b)                 Sx^-4x-2  a{x  +  2a) 

27.   ^-=^-       28.  a;2+i.       29.  m  +  2n.        30.  "  +  ^~^ 


x'f+y^  ab{a-c){c-b) 

31^   x^  +  a;y  +  y\  ^^         2{x  +  y)      .  33^    Oj^ 


34.    -^.      36.       8^^    .     36.        ^"""-^^      .     37.       ^^<^^  +  ^) 


a8-256  (x2-l)(x2-4)  (a-6)(a2+62) 

38.    ^(^^-1>.  39.   20xl-_34 

x4  +  x2  +  l  (3x-l)(2x  +  5)(4x  +  3) 

13  a 


40. 


(2a-3)(3a  +  4)(5a-2) 
§  151 ;  pages  120  to  124. 


2. 

10. 

13. 

8 
3* 

23. 

1 
4* 

35. 

-4. 

45. 

4 
5* 

3. 

-2. 

14. 

—  5. 

24. 

2. 

36. 

4. 

46. 

-2. 

4. 

3 

15. 

4 

25. 

5 

37. 

11 

47. 

1 

2 

3 

2 

6 

2 

6. 

3 

16. 

2 

26. 

2 

38. 

7 

48. 

_9 

5 

3 

7 

3 

2 

6. 

5 

7* 

17. 

6. 

27. 

3 
6' 

39. 

3 
5' 

49. 

1 
11* 

7. 

1 

2" 

18. 

—  0. 

30. 

11 
4* 

40. 

2 
17' 

50. 

19 
3 

8. 

4. 

19. 

-1. 
2 

31. 

-1. 

41. 

2. 

51. 

2 
3' 

9. 

5 

8* 

20. 

7. 

32. 

_4 
5 

42. 

19, 

9' 

52. 

6. 

10. 

4 
3* 

21. 

-1. 

33. 

1 
3' 

43. 

43 

7' 

53. 

2 
6' 

11. 

-1. 

22. 

-4. 

•34. 

1 
3* 

44. 

3. 

12  ALGEBRA. 


§  153;  pages  125,  126. 

2.  II-  9.  -2  a.  14.  1 

3  6  n 

3.  -5^.  9.  -?— .  15.  ^.                     21.    - 

a  a  —  b  b 

4.  -a.  10.  2(a-  &).  16.  -3a. 

5.  1^.  11.  m  +  n.  17.  ^. 

3  w  a 

6.  a-1.  12.  2^±^.  18.  _^.                24. 


a  +  i!) 

6 

a  +  6 

2 

3  ?7lJi 

ah 

a-b 
a-\-b 

m  —  \ 


2  36  2 

13.    12(a-5).         19.    2a -36.  25.    -6. 

■//* 

§  154;  page  126. 

2.  .09.  4.    5.  6.    — •  8.    .6.  10.   0. 

500 

3.  -4.  6.    -20.  7.    -.02.  9.    -1.4. 

§  155 ;  pages  127  to  135. 

2.    40.  3.    56.  4.    42.  5.    27,  18.  6.    32,  24. 

7.    A,  $40  ;  B,  $  48  ;  C,  $  36.     8.  Water,  288 ;  rail,  360  ;  carriage,  120. 

9.    A,  24;  B,  64.       10.    $25.        11.    $2.45.        14.    10  f.        16.    l/j. 

16.    15f  hours.         17.    If  minutes.         18.    48.         19.    82.        20.    79. 

21.   20.  22.    A,  24;  B,  48.  23.   \'  26.    35,  14. 

8 

27.    A,  30  miles  ;  B,  36  miles.  28.    107,  27.  29.   — • 

15 

30.  59.  31.    Horse,  $250;  carriage,  $175.  32.    6. 

33.  Horse,  $180;  carriage,  $280;  harness,  $30. 

34.  Express  train,  45  miles  an  hour ;  slow  train,  30  miles  an  hour. 

35.  A,  32  miles  ;  B,  25  miles.  36.    120. 

38.  38j2f  minutes  after  1.  39.    2,%^^  minutes  after  6. 

40.  ^\^j  minutes  after  4.  41.    lOi^f  minutes  after  5. 

42.  87.        43.    22J  miles.         44.    A,  3  days  ;  B,  6  days  ;  C,  8  days. 

45.  49Jj-  minutes  after  9.  46.    A,  $36  ;  B,  $32  ;  C,  $27. 

47.  10}f  minutes  after  8.     48.    45  minutes.     49.  A,  $1200  ;  B,  $900. 

60.  Longer  piece,  30  yards  ;  shorter,  24  yards.  61.    $  1840. 

52.  2l^j  and  54j-\  minutes  after  7.  63. 


ANSWERS.  18 

64.  Gold,  1540  oz.  ;  silver,  420  oz.  65.    $4725. 
56.    A,  4;  B,  5;  C,  6.            57.    2  p.m. 

58.  $  1250  ill  41  per  cent  bonds,  $  1750  in  3^  per  cent  bonds. 

69.  24  miles  an  hour.  60.    16j-\  minutes  after  10.  61.    7. 

62.  $18000.  63.    $2400.  64.    Gold,  57  oz.  ;  silver,  70  oz. 

65.  Fox,  180 ;  hound,  135.  66.    $5400. 

§  156  ;  pages  136, 137. 

an  am  ^     ^     am  —  amn  „   a  —  an 

2.    , 3.    A, years;  B,  years. 

m  -\-  n    m  -i-  n  m  —  n  m  —n 

4.    J?»iL_.        6.   ^ 6.    ^  -  ''''  dollars.       7.    ^-^^• 

m-\-  n  ab  +  be  +  ca  \  +  r  a  —  c 

8.    ^^  miles.    9.    -^.     10.      ^^'^^     dollars.    11.    lM«:zP). 
6  -t  c  b  -  a  100  +  ri  pr 

p«  6+16-1-1 

14.    A,  -«^  miles  ;  B,  -^^  miles.  15.   ""'''  +  &^  +  ^P  cents. 

m-h  n  m  -\-  n  a  -\-  b  +  c 

16.    First  kind,  ^li^^l^ ;  second  kind,  ^i^~J^. 


6  —  a  6  —  a 


amn 


«M  uimt  uii  a 

1  +  n  -|-  mn'   1  +  n  +  mn    1  +  n  -f-  wi» 

18    A    — ^-^l^^^l— •  B  ^  w»^P  o  2mnp 


mn  -\-  np  —  mp  mp  +  np  —  mn  mn  -\-  mp  —  np 

§  164;  page  141. 

3.    x  =  2.  3^^1  11.    x  =  -2.  14.    ^^_5. 

y  =  3.  2  2  2 

16.   x  =  4. 


6.   x  =  S. 


5  io    /..  — 2 


,  =  _5.  9.   .  =  -1  12.0.  =  -.  ,=-1 


6.   «  =  -!. 


16.    x  =  -  6. 


-4.  y  =  2.  y=_s. 

6  ^  17.   x  =  -3. 


,.   ._3.  10.   x  =  -|. 


4 


13.   x  =  -5. 


^3  ^4  ^*-    ^--^-  18.    x  =  9. 

^4*  ^6  «  =  4.  y  =  7. 


L4 

ALGEBRA. 

§165; 

page  142. 

2. 

3. 

X=:S. 

y  =  4. 
x  =  -4. 

8. 

x  =  l. 

-1- 

11. 

X  =  — 

2. 

14. 

2/ =-4. 

4. 
5. 

y=-i. 

x  =  2. 

y  =  6. 

x  =  5. 

y=-7. 

9. 

12. 

-1 

15. 
16. 

x  =  -5. 

y  =  \- 

6. 

7. 

x=-l. 
y  =  s. 
x=-S. 
y=-2. 

10. 

4 
-1- 

13. 

y  =  - 

3, 

2* 

17. 

y  =  -3. 

x  =  4. 
y=-b. 

§  166 ;  page  143. 


2.   x  =  2. 
2/ =  5. 

7. 

—I 

10. 

^-1- 

13. 

x  =  6. 
2/=-l. 

3.   x  =  4. 
2/ =-3. 

— 1 

y-l- 

14. 

x  =  -l. 
2/ =-5. 

4.   x  =  -6. 
2^  =-4. 

6.   x  =  l. 

2/  — 2. 

6.   x  =  -2. 

8. 

-1- 

11. 
12. 

2/ =  3. 

15. 
16. 

x  =  3. 
2/ =  2. 

-1 

9. 

x  =  -3.    • 
2/  =  l. 

-!• 

17. 

x  =  -4. 
2/ =  7. 

§  167 ;   pages 

144  to  146. 

2.    x  =  6. 

6. 

X 

=  -8. 

10. 

x  =  4. 

14. 

x  =  -5. 

2/ =-10. 

y 

=  5. 

2/ =-5. 

2/ =-7. 

3.   x  =  12. 

7. 

X 

=  -6. 

11. 

x  =  l. 

15. 

x=-7. 

2/  =  -  12. 

y 

=  -3. 

2/=-2. 

^  =  8. 

4.    x  =  -l. 

8. 

X 

=  3. 

12. 

x=-l. 

^=1- 

2/ =-5. 

y 

=  -5. 

2/  =  5- 

16. 

6.    x  =  4. 
2/ =-3. 

9. 

X 

y 

=  18. 
=  6. 

13. 

x  =  5. 
2/ =9. 

y-l- 

ANSWERS. 

11 

17. 

X  =  -  12. 

20. 

x  =  .8. 
?/=-  .07. 

-  ^=f- 

25.   x.lJ. 

18. 

x  =  5. 

21. 

x  =  2. 

y=-ll. 

^=-1- 

--!■ 

-1- 

24.    x  =  7. 

2 

19. 

x  =  -2. 

22. 

x  =  3. 

y  =  lO. 

26.    x  =  -10. 

y=-6. 

?/=-l. 

y  =  6. 

§168;  pages  147,  148. 

^  35 g  +  24  b  7.   x=-2a.  14.    x  =  a. 

'   ^  23       *  y=b.  y  =  b. 

^^14a^_186.  8.   x=-3m.  16.   x  =  a. 

^^  y=-2n.  y  =  a. 

3.    x=-^-i^.  g    ^^aa'(bc'-hb'c)  16.    x  =  wi^^. 
«^  +  62                    •         cc'(a'6  +  a6')  y  =  mn*. 

^      a^+&2  y=../L.^.;./N-  17.   x  =  5L±^ 


4.   x  = 


w'n 


cc'{a'b-\-ab') 


y  =  -\  6. 

»=  =  *•  r= 

y  =  —  6.  ^      m'p  —  mp' 


cia 
a2 

+  6) 
+  b'i 

c(a 
mn' 

-&) 
—  wi'n 

n'p 
mn' 

-np' 
-  w'n 

a 


?i^p  -  up'  10.    x=a.  _q-  & 

mn'  —  m'n                  y=  —  b.  ^~     5    * 

mp'  -  m'p                        or  n , 

y=^^^     J^'  11.   x=^.  18.   x  =  «  +  26- 


2  *"•    *-       2 


- dm  +  bn  y=---  ..      a-26 

0.    x  — ■ — •  ^         o  V  = • 

ad-{-bc  ^  ^2 

*    ^  _cm-an  12.   x=a'^-\-b. 

^--^dTbE'  y=a-b^.  19.  ^  =  4"^- 

6.   x  =  a  +  6.  13.   x=a(2a+6).  y  =  ^~^ 

y  =  a-b.  y=b(a-\-2b).  2 

§  169 ;  page  149. 

2.   x  =  -3.  5    ^^  a'^  +  ft^ 
y  =  5. 

3.  x  =  -.  y  = 

4 


7. 

x  =  -6. 

2/ =-2. 

8. 

x  =  3. 

2^  =  4. 

9. 

^=r 

--!• 

16  ALGEBRA. 


10.   x  =  a-\-h.  11.    X  =^. 

^      a+6  ^         3 


§  170 ;  pages  151  to  153. 


3. 

x  =  3. 
2/ =  2. 
i=-l. 

10. 

x  =  l. 
6 

17. 

22. 

w  =  6. 
a;  =-7. 
y  =  S. 

4. 

a;  =  -5. 

'  =  —s 

1 

z=-9. 

y=-4. 

11. 

x  =  -'S. 

^=3 

b-hc 

5. 

z  =  2. 

X  =  2. 
2/ =  5. 

2/ =  4. 

^=1 

18. 

a;  =  2. 

2/ =  4. 

23. 

z  =  -l. 

12. 

x  =  -5. 

z  =  Q. 

_        2 

6. 

x  =  -4. 

2/ =4. 
0=-3. 

19. 

x  =  '-. 

a  +  6 

y=-S. 
z=-6. 

13. 

a;=-  1. 
2/ =  6. 

4 

-1 

24. 

M  =  -5. 

x  =  4. 

7. 

x  =  -6. 
y^-7. 
z  =  S. 

14. 

z=-4. 
x  =  5. 

y  =  i. 
z  =  S. 

20. 

x  =  a. 

2/=-3. 
«=-2. 

8. 

x  =  -2. 
t,=-5. 
z=-S. 

15. 

x  =  -S. 
y=-b. 

2  =-7. 

y  =-  a^. 
z  =—  a^. 

25. 

w=10. 
x  =  2. 
2/ =  4. 

9. 

^4:- 

16. 

-\ 

21. 

a 

;s  =  6. 

y-l 

y-\ 

y=T 

26. 

x  =  6. 

1 

4 

Z=^' 

2/=-2. 

«  =  -. 

4 

^          5' 

c 

«  =-4. 

87. 

x  =  2. 

28.   x  =  6. 

29. 

X 

=  - 

12. 

2/  =  3. 

2/ =  14 

y 

=  - 

24. 

;?=-!. 

z=- 

12. 

z 

=  36. 

\n 

^_      2a&c 

2a&c 

z=  — 
at 

2abc 

JU. 

ab-\-ac—bc 

^~ah  +  bc- 

ac 

Q-i-bc-ab 

31. 


32. 


ANSWERS. 

5c  =  ab. 

33. 

x  =  a. 

y  -  be. 

y  =  l. 

z  =  ca. 

1 

X          2^^ 

a 

b  ->r  c-  a 

V-        ^'''' 

34. 

x  =  3. 

"      c-\-a-b 

2ab 

y=-l. 

a  +  b-  c 

z  =  6. 

17 


§  172 ;  pagres  155  to  164. 
3.    35,  24.      4.  20,  12.      6.  — •      6.  — •      7.  Apples,  ^1;  flour,  $3. 

C7  li7 

8.  A,  24 ;  B,  40.  9.  '26,  15.  10.  -|. 

11.  A,  35;  B,27.  12.    A,  15;  3,22^. 

13.  §630  in  4^  per  cent  stock,  $  810  in  3^  per  cent  stock. 

14.  Income  tax,  $28  ;  assessed  tax,  $36.  15.  A,  $60  ;  B,  $52. 
16.  $1.75,  $1.50.            17.  13,  17,  19             19.  84,  at  2^  cents  each. 

20.  45  cents  ;  15  oranges. 

21.  ??»«i«+Al  persons;  each  received  «M»L±i^  doUars. 

bm  —  an  bm  —  an 

22.  21  quarter-dollars,  13  dimes. 

23.  26 1  of  first  kind,  43 1  of  second  kind. 

24.  45  of  first  kind,  63  of  second  kind.  25.  A,  16 ;  B,  30  ;  C,  60. 
26.  32  for,  22  against.         28.  97.         29.  896.        30.  83.        31.  59. 
32.  4  from  the  first,  3  from  the  second.  33.  85  ft.,  64  ft. 
34.  A,  9  ;  B,  5.                       35.    467. 

86.   Express  train,  45  miles  an  hour ;  slow  train,  27  miles  an  hour. 
37.    A,  $72  ;  B,  $81 ;  C,  $63  ;  D,  $  180.        38.  First,  38;  second,  18. 

40.  Rate  of  crew  in  still  water,  ^^^  +  ^""^^  miles  an  hour ;  of  cun*ent, 

^»^=^  miles  an  hour.        ^'^'^ 
2mn 

41.  Going,  10, I  miles  an  hour ;  returning,  4V  miles  an  hour. 

42.  78.         43.  369.         44.  75  ft.,  54  ft.         45.  $ 375,  at  4  per  cent. 

46.    ^^  -  ^'^  dollars,  at  ^^^C^  "  ^>  per  cent.  47.  A,  15  ;  B,  21. 

m  —  n  bm  —  an 

48.   $  2000,  at  6  per  cent. 


18  ALGEBRA. 

60.  Rate  before  accident,  36  miles  an  hour  ;  distance  to  B  from  point 

of  detention,  90  miles.  51.    647. 

62.  A,  $6;  B,  $12;  C,  $8;  D,  $20.  53.  A,  .$13 ;  B,  $7 ;  C,  .$4. 
54.    Fore-wheel,  9  feet ;  hind-wheel,  15  feet. 

65.  A, days;  B, ; days:  C, days. 

mn-{-np—'mp  mp  +  np  —  mn  mn  +  mp  —  np 

56.    A,  8  ;  B,  12  ;  C,  24. 

67.  First,  $15000  at  4^  per  cent;    second,  $18000  at  3i  per  cent; 

third,  $  13000  at  5^  per  cent. 

68.  A, ^ hours  ;   B,  -^^  hours. 

6-f-c  —  a  a  —  b 

69.  Rate  of  crew  in  still  water,  9  miles  an  hour ;  of  current,  5  miles 

an  hour.  60.    Principal,  $5000;  time,  3  years. 

61.  A,  $55;  B,  $19;  C,  $7.  62.    12,  each  paid  $3. 

63.  Express  train,  40  miles  an  hour  ;  slow  train,  25  miles  an  hour. 

64.  A,  18  ;  B,  16.        65.    3  quarter-dollars,  8  dimes,  9  half-dimes. 

66.  30  of  3^  per  cent  stock,  20  of  4  per  cent  stock.        67.  A,  8  ;  B,  7. 


§  184 ;  pages  168,  169. 

3.   a;<3.         4.   x>i         5.    x<-.         6.   x>8.         7.   x<^' 
3  2  ^ 

8.  x>a-6.  9.  a;<l,   2/<4.  10.  ic>3,   y<2. 

11.   x>6  and  <9.       12.   7.       13.   18  or  19.        14.  38,  39,  or  40. 


§  187  ;  page  172. 

4.   x*  +  4x3  4-6x2-f  4x  + 1.  6.   4^4  -  4a3  +  17  a2  ~  8a  +  16. 

7.   25a;*  -  30x3  -  a:-2  +  6x  +  1.  8.   9x'^+2ix^+2Sx^+mx+i. 

9.  36  n6  +  12n4-60wHn2-10  71+25. 

11.  a*  -  8  a8&  +  22  a262  _  24  «63  4.  9  54. 

12.  4 x4  +  12 x^y  +  13x2y2  +  6 xy^  +  y^, 

13.  x6  +  12x5  +  36x4  -  14x3  -  84x2  +  49. 

14.  16  a^  -  40  a^x^  +  a^x^  +  30  a''-x^  +  9  x^^. 

17.  x6-2x5-x4  +  6x3-3x2-4x  +  4. 

18.  a6  +  4a5_2a*-20a3-7a2  +  24a+16. 

19.  4x6  -  20x5  +  41  x^  -  52  x3  +  46x2  -  24x  -|-  9. 


ANSWERS.  19 

§  188  ;   page  173. 

4.  x8  +  6x2  +  12x  +  8.  8.  210a3_i08a26  +  18a62-.63. 

5.  27a3-27a2  +  9a  -  1.  9.  125x3  +  150x22/+60x|/2+8y3. 

6.  m=5-12m2n+48mw2-64n3.     10.  64m3-144  w2w3+108mn6-27n». 

7.  x6  +  15x*  +  75x2  ^  126.  11.  27 x^ -  135x5+ 225 x*- 125x3. 

12.  64  xi2  +  240  x^yz^  +  300  xl^yH^  +  125  yH^. 

13.  8x3-84x5  +  294x^-343x9. 

14.  125  ai8  +  450  a^'^h^  +  540  a^h^^  +  216  ¥^. 

16.  a3  +  63  -  c3  +  3a26  -  3a2c  +  362^  -Zh^c  -\-Zd^a  +  3c26  _  6a6c. 

17.  x5  +  3x5  +  6x4+7x3  +  6x2  +  3x  +  l. 

18.  x8-y3+8^-3x2|/+6x25;  +  32/2x+6i/22  +  12  02^-12  22y_i2iry0. 

19.  a®  -  9  a^  +  24  a*  -  9  a3  -  24  a2  -  9  a  -  1. 

20.  8x6  +  12x5  -  30x4  -  35x3  +  45x2  +  27x  -  27. 

21.  27  -  108  X  +  171  x2  -  136  x3  +  57  x*  -  12  x^  +  x«. 

§  193  ;  page  176. 

24.    56.             25.    135.              26.    252.  27.  432.              28.    588. 

29.    24.             30.    105  a6c.       31.    462.  32.  45.                33.    12. 

34.   6.               35.    i^6.              36,    28.  37.  a3  +  4  a2  +  ^  -  6. 

§  195  ;  pages  178,  179. 


3. 

4. 
5. 
6. 

7. 

2x2  + x  +  1. 
l-3a  +  a2._ 
.3x2_4x-2. 
2x2  +  5x-7. 
a  —  h  —  c. 

10.  3x  + 

11.  7m2 

12.  3a2- 

13.  5x2- 

5y-4^. 

—  nin  —  4  ??2. 
-5a +  4. 

-  2  xy  -  3  y2. 

16.  m  +  4-1. 

m 

17.  1  -  X  +  x2  -  x\ 

18.  x3-4x2-2x-.'^ 

19.  X  y  ^y\ 

8. 
9. 

2  a3  +  3  a2  -  1. 
x3-2xa2  +  5rt3. 

14.  4m2 

15.  3a2- 

+  mx2 
-2a6 

-3x*. 

-5  62. 

2      2x 

21. 

2a3  +  3a26  +  4a62_5  63. 

26. 

1+a 

-f-f-- 

22. 

a^     ah     62 

2       3       4* 

27. 

1-^. 
2 

x2     a;3 

8      16      "*' 

23. 

3  x3  -  2  x2y  -  a;y2 

+  4^3. 

28. 

J      3a 

2 

;     9  a2     27  a' 
8         16 

24. 

4     X     2  x2 
3     a      a2 

29. 

x  +  §- 

X 

-    ^   +  2^  +  .... 
2x3^2x5 

25. 

l  +  2x-2x2  +  4.x8+  .... 

30. 

2a-- 

2 

6         62          6'* 
la     16  a3     64  a5 

20  ALGEBRA. 

§  199 ;  pages  182,  183. 


1. 

65. 

10. 

3581. 

20. 

3.6055. 

30. 

.8660 

2. 

148. 

11. 

274.9. 

21. 

6.9282. 

31. 

.7453 

3. 

713. 

12. 

.4027. 

22. 

8.0436. 

32. 

1.148. 

4. 

8.07. 

13. 

51.04. 

23. 

.44721. 

33. 

.7071. 

6. 

.396. 

14. 

.07906. 

24. 

.23664. 

34. 

.7745. 

6. 

.254. 

15. 

9.318. 

25. 

.62449. 

35. 

.9354. 

7. 

62.9. 

n. 

2.6457. 

26. 

.094868. 

36. 

.6373. 

8. 

9.82. 

18. 

2.8284. 

27. 

.027202. 

37. 

1.035. 

9. 

.0567. 

19. 

3.1622. 

39. 

28. 

.6085. 

2.9265. 

38. 

1.258. 

§  201 ;  pages  185,  186. 

7.  x'^-2x-l.  11.    a2  -  3 a  -  2.  14.    x2  +  2 xy  +  4 y^. 

8.  2a2  +  3a  +  l.  12.    2x'^-bx-\-2.  15.   ^-1+1 

3  X 

9.  Sy^  +  y-  2.  13.    Sa^-2ah-\-  b^. 

§  206  ;   pages  189,  190. 

1.  27.  6.    9.5.  11.    .0481.  16.    1.442.  21.    .7413. 

2.  53.  7.    .608.  12.    92.4.  17.    1.912.  22.    .7631. 

3.  3.9.  8.    3.59.  13.    7.63.  18.    2.087.  23.    .7368. 

4.  .85.  9.    806.  14.    697.  19.    .2714. 

5.  136.  10.    57.2.  15.    .1048.  20.    .8549. 

§  207  ;  page  190. 

1.    Sa  +  2b'^.        2.    \-Zx-x\         Z,    2a'^-a-  2.        4.    x^  +  y2. 
5.    a -2.  6.    21.4.  7.    .46. 

§  217  ;  pages  195,  196. 

8.  wi      9.  A^       10.  6w"i       11.   7a~^i       12.  Qah^.       15.  ax~'^ 
17.  a-h.     18.  8a;-2  +  27.     19.  8a-2-18a-i-47-15a.     20.  x-^-\Q 

1112  7  fi     _4  _5  4  3    _1 

21.  ar^  +  x^ys+y'J.       22.  m^n-^— 4w^n  ^-\-Qmn  3— 4  m'n'^+w'n  ^ 
23.  a-%-^  -  3  a-^b-'  +  a-'6-9.  24.  2  m~^  +  4  mThr^  +  18  n-* 

25.  4a^6-2-17a^?)2+16a-f66.  26.   ]^m^x~'^ -20m^x^+2m~'^x, 


ANSWERS. 


21 


§  218 ;  pages  196,  197. 

1.  nk         9.  3a:"K         11.  a*  -  a^ft^ +&^. 


bK         6.  2-a;'s\ 

-^  -L^  13.  a;2  _  2  -f  x-\  14.  a^  -  2  a^  +  1 


_3  _1  _1 

a  ^'  +  a  -^  +a  *  +  1. 

^^•-^         16.  m-2-2?>ri  +  l-2wi.        17.  3a;V+a;2y+a-. 

19.  aV5-2-3a"26i 

2     3 


_3  4  2     3 

20.  m^x  ^  +  2m^  +  w^x^. 


s 
a;^. 


wi' 


§  220  ;  page  198. 

11.  a^  13.  c' 

12.  a;"^.  14.  a" 


16.  m~^. 

m 

16.  X". 


2.    125. 


243. 
256. 

27. 


§  221 :  page  198. 


«.i. 


8.  128. 

9.  49. 


11.  -  128. 

12.  32. 

13.  625. 


15.  -1024. 

16.  81. 

17.  -I 


§  223 ;   page3  199,  200. 
2a*+l-5a"^.     9.  3a;"3-2x"5-f  1.     10.  Jb-^-iab-^-Sah-\ 


+  2x^-3x1  17.    a 


18.    tt^^-sn,  19,   a;"'-'. 


X 

2(x 


-1.        21.    X""*.        22.    a'.        23.    -•       24.    z^'^"*.        25.    ^i^- 

8  1  -  rt 


'  +  yh 


x^  -y- 


x"  +  2. 


27.    a"  +  1  +  a 


30. 


2  xy 

x^  +  y^ 


28. 


31. 


2(x  +  y)     . 

a;-2/-(x  -  y) 

2a  +  16a^ftf 
a  -  86 


12. 
13. 


VUab^. 
V5  xy^ 


§  228 ;  page  202. 

14.  V2ahn. 

15.  V3  m^n^ 


16.  \/2x3m2. 

17.  v'^^s^ 


22 


ALGEBRA. 


§  229 ;  pages  202,  203. 
19.   6 abWS ab'^  +  2 a^b.  22.    (x-\-S)V6x. 


20.  3x?/v^5xV-4  2/*. 

21.  (a  -  2  6)  Va  +  2  &. 

27.    12  V6.  29.    42\/2. 


28.   5\/l05. 


30.    75  Vs. 
35.    12^. 


23.  (^a-2b)VSab. 

24.  (x  -  3)  \/x2  +  7  a;  +  10. 

31.  28\/42.  33.    7\/l2. 

32.  5\/9.  34.    14^28. 


36.   315a6>/l5a&. 


2.  i\/6. 

3.  fV5. 
12.    ^v^. 

16.    —  \/42a. 
6a 


19.    |^V3^. 


§  230 ;   pages  203,  204. 
4.    i\/T5  6.    j\\/^5.  8.    -|Vl2. 


5.    |V2. 


7.    |\/3i. 


9.    aV2. 


13.    |\/10. 
17. 


10X2 

20.    J-V98«2. 
7a 


14.    iv/8t, 
V30^. 


Va^  -  b\ 


X  +  2 


10.  iv^. 

11.  1^5. 
15.    ^\/l8. 

18.     ^y/W^. 

6cd3 

21.    -Lv/20^. 
4y 

V2x. 


14.    vT 


15. 


§  231 ;  page  204 
Vx2"^=n.         16. 


\a  -  b 
a  +  b 


17.  J(£.^ 

>!     .X2  +  1 


V2. 

3/ 


§  233 ;  pages  205,  206. 

3.    7\/3.      4.  4\/2.      5.   -2V5.       6.    5^2.       7.    3v/3. 
9.    5\/3.        10.    \/7-2Vn.        11.    V-v/2.        12.    |V0.        13.    1^6 
14.    0.  15.    -i^VTO.  16.    2^9-3^5.  17.    -i>/l5 

18.    -a2&2V2^.        19.   lOm^v'Im^^.        20.    (5  a  -  4x2)  V2  a^  -  3x 
21.    fVIi.  22.    v^.  23.    6^3-2^6.  24.    -Sv^B 

25.    7V2-5\/5.  26.    4x\/6^.  27.    2  62  VTo^  -  3  a  ^76 

29.    ^§\/30-fVlO.  30.    (7x-l)V5x 

31.    13?/\/3.  32.       ^ 


28.    ^V3-V6 


a  —  b 


-J a?-  -  bK 


ANSWERS.  .       23 

§  234  ;  page  207. 

2.  \^,   V^.  3.     v^,   '</\K  5.    \/l28,   v^lii. 
6.   \/256,   v^2l6.                   8-    y/sTa'^   y/W^   Vm7^. 

9.  '^64,    \/512,   v/l6».  10.    v^l  -  :i  x -\- S  x^  -  xr^,   '^F+l^T^. 

11.  v^a3  +  3  a-'6  +  3  aft^  4.  53^  \/a*  -  4  a^ft  +  6  a-^62  _  4  a^a  _,_  54. 

12.  \/3.  13.    VE.  14.    ^4.  15.   V6>v/l4>v^l75. 
16.  '-(^253  >  V3  >  v/T5.                     17.    V3  >  Vo  >  v/?. 

§  235 ;  pages  208  to  210. 

4.  12.       5.  6  a.       6.  6V7.       7.  5V3U.       8.  110.       9.  10 ay/^Tbc. 

10.  12.  11.  3v'35.  12.  Gv/55.  13.  |Vr5.  14.  3v^. 

15.  2^^.  16.  P>x^^.  17.  2\/486.  18.    v'SOO^^^. 

19.  5v^5.  20.  2  6v^l6a56cs.  21.  3v^.  22.  2v^27. 

23.  3v^.  24.  i</m.  25.  ^v'TsS.  26.   v^^sfc^^. 

27.  2\/3.  28.  2v/l08.  29.    v^.  32.    2  +  7\/3. 

33.  12  X  -  6  +  16\/2l)-.     34.  202  -  GSVlO.     35.  54  a  -  55  6  +  IJ9  Vab. 

36.  165  +  18  v^  +  35v^l00.  37.  «-46  +  9c-  6Vac. 

38.  22 a: +  2  -  23v'.t2  -  1.         39.    -2-2\/l5.        40.   -  72  +  33\/3. 

41.  8  +  30vT5.     42.  140-48\/lO.     43.  -484-54\/6+12\/l0  +  60Vl6. 

44.  _  47  _  2VT5  4-  25>/6.  45.  61  +  24  V5.  46.  37  -  20\/3. 

47.  168  -  96\/3.        48.  665  +  70V70.        49.  5  a  -  4  +  2\/iU^-Sa. 

50.  13a; +  5y-  12  Vx^  -  y\        51.   -31.        52.  28.        53.  4  -  21  a;. 

54.  2  6.  65.  3-46^. 

§  236;  page  211. 

3.  2V3.         4.  |V5.         5.  ^>/7.         6.  3^3.         7.  h         8.    i\/225. 

3 

9.  3.  10.    J-v'T62^.  11.  2V'2.  12.    \/2.  13.  a/-^- 

3  a  >'5c 


14.    ^^-.        15.  I       16.  AVI5.         17.  ^>/ J^.       18.  ,^3^^^. 

19.    vWoF.         20.  ^v^TeO.  21.  a/--        22.    ^^^-        23.    ^. 

'2  '  2y 

24.    ^.  25.   J- ^18^.  26.   ^^^8. 


24  ALGEBRA. 

§  237  ;  page  212. 

6.  18\/2.  8.   av^.  10.  3v^.  12.  ^Om^x^STi. 

7.  S2  a'^b^Vab.      9.  5\/2^.  11.  2v^.  14.  2?/v'a^. 

§  238;  page  213. 

8.  \^.         10.   \/l&2xy^         11.    \^.         13.    v^.  14.   </dV\ 

§  239;  page  213. 

2  V6^  4    ^  6     ^^^  8    ^ 
■3'                      '5'                               2*  '2' 

3  ^^7^  5    4\/9^  v^  g    3v/4^3^ 

'    7  a62  '  •       3     '  *     3  '  *       2  a:y^     ' 

§  240;  pages  214,  215. 

3    9-SVE  g    53  +  12\/T0  j3    2  g^  _  ?y2  _  2ffVa2  _  52 

'  2       *  '  37         *  *  ?)2 

^        5>/3+  10       9    22VT5-85  ^^    l  _  Vi~I~^ 

2        '        *  5  "  '  a  " 

g  a  +  b^  +  2h y/a  -^  a;  —  4  —  Va;  —  2  --  x  +  Vx'^  —  y'^ 
a  —  b'^  X  —  4  y 

g  a;  +  y-2\/xy  ^^  &  -2a-2\/q2_qft  ^^  Vg*  -  x^  -  a^ 
X  —  y  b  x^ 

7  9  +  4V3  -2  1  +4a;\/l  -4x-^  -„  I4x  -  10  +  llV^^^^ 
3       *  •  8a;2-l         '  5X-13 

§  241;  page  215. 

2.  .949.  4.  .535.  6.    -4.560.  8.  .268.  10.   -.330. 

3.  2.224.         5.  .684.  7.  4.442.  9.  13.354 

§247;  page  218. 

3.  V7+2.  8.  2\/7-V2.    13.  3\/5-l.  18.  2\/5-\/l5. 

4.  3-2\/2.         9.  3-V3.  14.  5+VTo.  19.  SVS+VlO. 
6.  4\/3+l.        10.    V6+\/5.      15.  5V'2+VG.       20.  6\/2-V3. 

6.  2V3+\/7.    11.  4+>/T0.        16.  3\/3-2V2.     21.   V^+T+V^^. 

7.  2\/6-2.        12.    vTl-3.        17.  4\/2-\/5.       22.   Vo^-Vft. 


ANSWERS. 


25 


§  251;  page  219. 

2.  9V^^.  3.  WS  v^^.  4.   \/2  \/^. 

6.   (a;  +  1/ +  i2)  V^n.  7.  0.  8.  SV^HT. 

10.   y/lV^.  11.  (l-a:)V^^. 


5.  bV^^. 


4. 
10. 
13. 
16.' 
20. 
23. 

27. 
31. 
34. 


§  252;  pages  220,  221. 

-14.  6.  12  a2.  6.  -2>/l5.  7.  -Va6.  8.  18.  9.  -60. 
26-7V^.              11.  66-33  V^^.                12.   -  61  + 18VI5. 

-  8a  +  18  ft.  14.  -  xyzV^^.  15.  48\/2  V^H". 
-8V30-17>/T5.  17.2.  18.480.  19.  -  v^lO. 
-2  +  2>/^^.             21.    -74-40>/3.  22.   11 -8\/^^. 

-  30  +  12\/6.           24.  x2  +  y            25.  61.  26.    -  9a  +  4 6. 

^  1  _  v^^ 


-  50.  28. 

73  +  40\/3 

23 

_  10  -  9\/^3. 


12 


32. 


51  -20\/l5 
33 


2. 


§  253;  page  222. 


6.    -V^. 

6. 3>/^n". 


'•a/^- 


8. 


Va 


30.    V-1. 
33.    -2  +  2V'3T. 


9.   V5.  11.   V3. 

10.   -Vs.         12.  V2. 


6. 


2 
3' 

16. 

25 
36* 
1 
6* 


9. 


9 


20 

10.  -2. 

11.  2. 
5 


12. 
13. 
H. 


§  254;  page  223. 
5a 


15.  4. 
25 


16. 
17. 
18. 


144 
16* 


19.  - 


).  1. 


21. 


4 

4  62 

10  g 
3 


25 


26. 


7a 
8  ' 


ab 


27. 


28. 


9a 
16* 


64 


5a 
4  * 

71 
120* 


31.  6. 


26 

ALGEBRA. 

§  256 

;  page  225. 

3. 

±3. 

7. 

±5.           11. 

±6.           15.   diSv" 

-1.  19.   ±\ 

4. 

4 

8. 

±T-      ^'^• 

-H-  --!• 

20.   ±(«-6). 

5. 

±\/3. 

9. 

4--   "• 

±4.       n.  ±8. 

21.   ±i>/l5. 

6. 

±6. 

10. 

±  2.          14. 
§  259 

±2.          18.   ±1. 
;  pagre  227. 

-   =^1- 

3. 

1,   -7. 

7.  5,   -  6. 

u.1,-5. 

15.  6,-1 

4. 

8,   -4. 

-•!• 

n.  .  |. 

18         1          ^. 

5. 

-2,    - 

9. 

9.1,-1, 

"•II- 

6. 

10,  3. 

10. -|,  -5. 

5  260 

;  page  229. 

• 

3. 

I'-- 

.    -1,    -. 

■    --l-l- 

in     1          1 

4. 

o 

5 
4* 

'•II- 

-  -\  -\ 

16.   i,    -1. 
9        3 

5. 

''!• 

-hT\^ 

^--\-\ 

6. 

1 

2'       4* 

10.  ^,  -|. 

u.  .  -I 

§  262 ;  pages  230,  231. 

3.10,    -3.  7.    -1.    -|.       11.    -f,    -\  15.5,?. 

S,,,    -|.         12.   I    -|.  16.  3,   I 

9.1,  I  13.1,-4.  n.|,-f. 

^»-l-i-    ^*--l-^-  "--i'-l- 


4. 

^'-1- 

5. 

y-  -- 

6. 

1         3 

4'        2' 

ANSWERS. 

2 

§ 

263;  pagres  231  to  233. 

1. 

«'-!• 

10. 

3, 

4 

5' 

20. 

-4,    -7. 

30. 

'i'- 

2. 

1- 

11. 

1, 

7 
18 

21. 

1         1 

~9'    ~8' 

31. 

-1,    -2. 

3. 

3    1 

12. 

2, 

1 
3* 

22. 

-3,    -4. 

32. 

^'1 

4. 

-!•  -- 

13. 

3, 

1 
7* 

23. 

1  1 

2  I 

33. 

-1,    -3. 

5. 

!■-■ 

14. 

1, 

2 
3* 

24. 

4,    -1. 

34. 

^'-t- 

6. 

U,    _3. 
o 

15. 

2, 

-  1. 

25. 

-3,    -4. 

35. 

1        2 

6'       7' 

7. 

^'1 

16. 

26 

,    2. 

26. 

^'k 

36. 

8         18 

5'  ~r 

8. 

5,    -6. 

17. 

6, 

-3. 

27. 

?•-• 

37. 

^-• 

9. 

0±V3. 

18. 

119^  7. 

28. 

2         23 
'        12* 

38. 

!•-■ 

19. 

3, 

-13. 

29. 

^'  -f 

r 

§  264 ;  pages  234,  235. 

3.   2a  ±3  6.         4.    1,    -2m -1.         5.    a,    -1.  6.    -  b,    -a 

7.    771^   m3.        8.    -,    --.       9.    3a-f5,    -a  +  7.  10.    1,   -^ 

c         a  rt  —  0 


11.    _«^,    _6.              12.    -«J:V«^^^.  13.    a  +  6,   «_±-^ 

a  +  6                                          a  2 

14.    2  a,    -a.                  16.    a,    -^^^tl.  le.    a  -  2  ?>,    -85 

2a 

,-     a  — 6     a  +  26               m     r  ^     13  a  -«     2  —  a         o^      1 

17.    — - — »   — 18.    5a,    — —  19.    — - — ,    —  2a-l 

A      ,        o                                          o  2   ' 

20.    (3a  +  6)2,    -(3a-6)-2.      21.  ^^±^,   ^^.  22.  a-2  6,    -2a  +  b 

a-b    a+6 


23.    3  a,   -^.       24. 

a- 

-c,  -6  +  c.     25.  42  a, 

2  a. 

26.      ^r,      2  m. 
3 

27.   3  a,    -4  a. 

28.    1,   ^-«. 
a  -  6 

a^  +  l     a^-l 

—    a2_i'    ^2^.1 

on    a-b        c 

31.    a +  6,     2«^ 

a+  b 

3„    ,     6  +  c-2a 

oU.               ,               • 

c        a  —  0 

^^-    ^'   c  +  a-26 

28  ALGEBRA. 

§  265  ;   page  236. 

3.  I    -2.  T.    1,    -|.  U.    -I,    -5.      15.    -I  1. 

4.  9,    -4.  ..   4,    -|.  12.   1,    -|.  16.    -!,    |. 

5.  -6,    -8.         9.   3,   -.  13.    -,   -. 

4  5    5 

6.  2,   ?.  10.    -i.  -^.       14.    -^,    -I- 

5  14*      2  3         4 

§  267 ;  pages  237,  238. 

6.    -7,  4.        6.    5,    9.        7.    -8,    -3.        8.    12,    -6.        9.    0,   ^. 

5 

10.    0,    -8.         11.    0,    ±?.        13.    0,    ^,    -?.        15.    3,    -^,    -4. 
4  4         3  2 

16.    0,   2drV2l.  17.    ±3a,    ?,    -a.  18.    -1,    ^  =^  ^^^-^. 

2  2 

19.    3,  :z3±3VH3.      20.   ±3,   ±?V3i.       21.  i«,   -2ad:2aV-3 

2  2        2  3  3 

22.    -|    5  ±  5  V^Ts  23.    i2,    liV^S,    _  1  ±  V^s. 

24.    1,    ±\/^.  25.    0,   |.  26.    ±5,   ^.  27.    ± -,    - -• 

2  5  2         2 

28.    -f,    ±3\/^2.  29.   0,   i^.  30.   0,   4a -4. 

4  5 

§  268 ;  pages  239  to  242. 

3.  21,  at  $6  per  barrel.     4.  11  and  7.     5.  9  and  16 :  or,  -^  and  -^. 

2  2 

6.  3,  4,  5.  7.  16  and  4  ;  or,  25  and  -  5.  8.  6  and  2. 

9.  1,  2,  3,  4  ;  or,  5,  6,  7,  8.  10.  18,  at  $6  per  barrel.  11.  21. 

12.  $40.  13.4,5,6.  14.  6  miles  an  hour.  15.300. 

16.  27  and  36  miles  an  hour.        17.  18  rods,  12  rods.        18.  20  cents. 

19.  $  75  or  $  25.  20.  9  miles  an  hour^ 

21.  Area  of  picture,  25  sq.  in.  ;  width  of  frame,  4  in. 

22.  Fore-wheel,  12  ft.  ;  hind-wheel,  16  ft. 

23.  Larger,  6  hours  ;  smaller,  10  hours.  24.  22.  25.  $  3000. 
26.  5.                    27.  5.                    28.  8.                    29.  4. 

30.    12100  and  1225  sq.  ft.  ;  or,  8836  and  4489  sq.  ft. 

M.    6.        32.   136  or  68  miles.        33.  72  miles.        34.  80,  at  $  60  each. 


ANSWERS.  29 
§  270 ;  paeres  244,  245. 

4.    ±3,   ±2V3.           5.  ^,  'f          6.  4,   </l2i.  7.   -1,  -|- 

8.    I  -1           9.    -243,  263.           iq.  |  _  ^|.  H.  729,  1- 

22.    1,  f?^t.            23.   «,  ^^            24.    16,  9.  25.   i^^. 

4    \6/                             a  8 


§  271 ;  pages  246,  247. 

4.    5,  -  3,  3,  -  1.  6.  3,  -  7,  1,  -  5.  6.  6,  -  1,  4,  1. 

7.    ±3,  ±3v^.     8.  6,  ^-     9.   1,  1±2VT5.     10.  0,  -2,  -1±2\/^. 
3 


jj     ^  1  3j:V-503      12.4,-1.     13.-2,-^.     14.-2,-5,-3,-4. 
2  4  8 

«.,,_1.3±^.  16.2,-1,,,1.  n.|,-2,|-,. 


18.   2,  l,9±vm.  jg    aj,Vai-4ft».     ^   a^SbVU,  a±2bV2b. 

4         8  2 

21.    3,  -1,  2±^.  22.  ^±:^.  9±:^^S1§.      23.  7,  -??,  3±iv^. 

2  2                6                                5          6 

24.    3,   --^S?.  25.    5  a,   -7  a,  a,   -3  a. 
'  128 


§  276 ;  pagre  250. 

2.    X  =  3,  y  =  ±  6  ;  or,  a;  =  -  3,  y  =  ±  5. 

4.   a;  =  2\/3,  y  =  ±2>/2;  or,  x  =  -2\/3,  y  =  ±2\/2. 

6.    X  =  2  a  -  6,  y  =  ±  (2  &  +  a) ;  or,  x=-2a  +  b,  y  =  ±(2b+a). 


30  ALGEBRA. 

§  277  ;   pages  250,  251. 

Note.  —  In  this,  and  the  three  following  sections,  the  answers  are 
arranged  in  the  order  in  which  they  are  to  be  taken  ;  thus,  in  Ex.  2, 
the  value  x  =  2  is  to  be  taken  with  y  =  3,  and  x  =  10  with  y  =  —  IS. 


2. 

x  =  2,  10. 

7.   X 

=  6,   1. 

12. 

.=-3.-f. 
.  =  .>. 

3. 

y  =  S,-  13. 

x  =  6,   -9. 
y=-9,  6. 

y 

8.    X 

y 

=  1,  6. 

=  a  +  l,   -a. 
=  a,   —  a  —  1. 

13. 

4. 

x  =  S,   -7. 
y  =  7,   -8. 

a;  =  10,   -3. 
y  =  17,  4. 

9.    X 

=  8,-3. 

14. 

5. 

10.  X 

y 

11.  X 

=  a  +  6,  a  -  6. 
=  a  -  6,  a  +  6. 

=  5,   -3. 

15. 

.  =  3.-?I. 

6. 

x=2,   -5. 
t/  =  5,   -2. 

y 

=-•!• 

.  =  12,  -f 

§  278  ;  page  253. 

4. 

a:  =  8,  6. 

9. 

X  =  5,  2. 

14. 

,   x  =  8,   -2. 

y  =  6,  8. 

y=-2,  -5. 

y=-2,  8. 

5. 

ic=l,  -10. 

10. 

x  =  -l,   -6. 

15. 

x  =  6,   -9. 

2/ =-10,  1. 

|/=-6,   -1. 

y  =  9,   -  6. 

6. 

x  =  4,   -3. 

11. 

x  =  6,  -7. 

16. 

x  =  4,  17. 

y  =  3,   -4. 

y=-7,  5. 

2/ =-17,    -4. 

7. 

a;  =  5,   -9. 

12. 

x  =  2,   -16. 

17. 

x  =  ±7,    il3. 

y  =  9,  -5. 

y  =  16,   -2. 

?/=Tl3,    T7. 

8. 

x=±Q,   ±2. 

13. 

X  =  4,  20. 

18. 

x  =  2,   -7. 

y=±2,  ±6. 

19. 

2/ =-20,    -4. 
a;  =  _6,   -25. 
y  =  25,  6. 

2/ =-7,  2. 

§  279  ;  page  254, 

x  =  ±4:,   ±|\/2.  3.    x  =  ±2,    ±fV2. 

?/=il,    TfV2.  !/=T5,    T|V2. 


ANSWERS. 


31 


4.  x  =  ±3,  ±4\/3. 
y=±6,  T5V3. 

5.  x=±4,  ±1. 

6.  x=±6,  ±i3tv/::r3. 
y=T4,  ±-V^V^=^. 

7.  a;  =±4,  ±  fV7. 
y=i3,  T|V7. 


8.  x=±2,  ±  j\v/-  13. 
2/=Tl,  ±3-VV-13. 

9.  a;  =  ±5,  ±  fV-  10. 
y=±l,  if^V^n^. 

10.  x  =  ±l,  ±fv^. 
y=T7,  ±fV77. 

11.  a;  =±2,  ±T%V3. 
2/ =±5,  TffVS. 


6.  X 

y 

7.  X 

y 

10.    X 

y 

13.    a: 

y 

16.     X: 

y- 

19.   X 

y 

23.   X 

y 

26.   X 

y 

28.    X 


=  ±4,   i^V^Ts. 

=  ±3,   T^^^. 

=  6,   -4. 

=  ±4,   ±iV46. 


§  280  ;  pagres  257,  258. 

6.    X  =  4,   -  3,   -  1  ±  >/l3. 


y  =  3,   -  4,  1  ±  Vl3. 
8.  x  =  ±l,  ±^.  9.  x  =  3,  6. 


11.  x  =  8,  11. 
y=-ll,  -8. 


4,2,8,--.       14.  x  =  2,  -4.       16.  x  =  2, 

'22 


12.  X  =  3,  9. 
y  =  9,  3. 


3,  8,  -  6,  16. 


y  =  4, 


y=-l,2, 


2 


±2,  ±V^.        17.  x=r3,  -6,  -,  --. 


18.  x=-5,  - 


55 


=  :fl,   ±2V^. 

=  a±l. 

=  «  T  1.  y 


.       9  9 

''  ~4'  i' 


y= 


-6,  -^. 

7 


20.  ^=\\'       21.  x=.6,  -6 


1   1 
4'  3* 


22.  x  =  a±h 
y  =  ±  3,  ±  3.  y  =  aTh. 


:2a-3,  1^.  24.  x=±3,  ±1.      25.  x=3a+2,2a-3. 

lo 


:3a-2, 


126  a -169 


26 

=  ±2,  ±Hv/-31. 
=  T2,  i}fV-31. 

b±y/-  151 
2 


y=±l,  ±3.  2/=2a'-3,3a+2.  • 

27.  x  =  ±(2a-6),  ±(a-26). 
y=±C«-26),  ±(2a-6). 


=  3,2, 


x  =  2, 


10  5±Vl93 


:-2,  -3,ZLi±^^EMI. 


3  4 

63  ±  3\/l93 
■  4 


21, 


32  ALGEBRA. 

30.   x=27,  -8.     31.  x=2,  -1.     32.  x=a  +  l,  -a.     33.  x=2,  12. 

y=8,  -27.  y=-l,2.  y  =  a,  -a-1.         y=-3, -i-- 

72 


34. 

J            4                                                       o 

y=||,o.                          ^  =  2'  |-                       ^  =  26,  -6. 

37. 

x=2,  -1,  l±:^Illl.     38.  a;=0,2,  ±V2.     39.  x=±3,  ±V-7. 

y  =  l,  -2,  Ill±^3dl.        ^=0,  2,  2T  V2.         i/=2,  6. 

40. 

a;=±l,±2.    41.    x=3, -1, -1, -2.     42.  x=3,4,  -6±>/i3. 

y=±|±|            y=l,  -3,2,  1.                 2/=-4,  -3,6±Vi3. 

43. 

a;  =  -2,  -4.              44.  x  =  3,  -  1,  2,  -  3.              45.  x  =  2,  1. 

y=_4,  -2.                    y=-],3,  -3,  2.                     y  =  1,  2. 

-^-•-V^— '•-!-=-• -i- 

§  281 ;  pages  258  to  260. 

1.    6,  ±  4  ;  or,  -  6,  ±  4.        2.    ±  5,  ±  3  ;  or,  ±  SV^  T  SV^^. 

3.    18  rods,  9  rods.  4.    7,  5  ;  or,  -  .5,  -  7.  6.    5,  2. 

6.    Cow,  l$70  ;  sheep,  $40.      7.  32  or  23.      8.  9,  4.       9.  -  or  :=^. 

-  8         22 

10.  24  in.,  16  in. 

11.  Rate  of  crew  in  still  water  6  miles  an  hour,  of  stream  3  miles  an 

hour;    or,  rate  of  crew  in  still  water  -'/  miles  an  hour,  of 
stream  |  miles  an  hour. 

12.  Length  30  rods,  width  12  rods  ;  or,  length  60  rods,  width  6  rods. 

13.  60  ;  A  gives  to  each  ^  3.     14.  A,  6  hours  ;  B,  3  hours  ;  C,  2  hours. 

15.  Length  32  rods,  width  30  rods.         

16.  6  and  4;  -  4  and  -  6 ;  or,  l±^  and   -^±/^K 

17.  A's  rate  of  walking,  3  miles  an  hour  ;  distance  12  miles. 

18.  A,  4  hours;  B,  8  hours  ;  C,  12  hours. 

19.  1  and  3  ;  or,  2  +  Zy/^^  and  2  -  Zy/'^. 


ANSWERS.  33 

§  283;   pagres  262,  263. 

2.  a:2- 15x  +  54  =  0.  8.    8x2+17x  =  0. 

3.  a;2  +  x-6  =  0.  9.   SGx^  +  77x  +  40  =  0. 

4.  3x2- ic -2  =  0.  10.   a;2  +  (2  6-3a)x4-2a2-5a6-3  62=o. 

5.  2  x"2  +  19  X  +  44  =  0.  11.    052  _  2  ax  +  a2  -  9  77i-  =  0. 

6.  30x2 -31x  + 5  =  0.  12.    x2-6x-89  =  0. 

7.  28x2-x-15  =  0.  13.   4x2  +  4xVa  +  a  -  6  =  0. 

§  285;   pages  264,  265. 

6.  (3x-2)(x  +  3).  19.   (9-4x)(5  +  3x). 

7.  (5x  +  8)(x  +  2).  20.   (7-2x)(6  +  5x). 

8.  (2x-3)(3x-l).  21.   (6x-5)(4x-  1). 

9.  (3x-4)(5x  +  2).  22.   (4x  +  5)(2  x  +  7). 

10.  (5-3x)(4  +  x).  23.   (3x-4y)(7x  +  62/). 

11.  (5-3x)(7  +  2x).  24.   (7 x  -  5ab)(x -\- Oab). 

12.  (6-x)(2  +  6x).  26.   (x  -  3y  +  4)(x  +  4?/ +  3). 

13.  (x-7a)(3x  +  4a).  27.   (x  -  2?/ -  l)(x  +  ?/ +  2). 

14.  (3x-7m)(2x-3w).  28.   (x -2y  +  4)(x  +  2y  -  1). 

15.  (7x  +  2)(2x  +  3).  29.   (2x  -  ?/ +  3)(x  +  4y  -  1). 

16.  (3x-2)(6x-l).  30.   (a-26-2)(3a4-6-l). 

17.  (1  -  4x)(5  +  X).  31.  (3y  -  2  -  x)(3y  -  3  +  4x). 

18.  (9x4-2)(2x  +  3).  32.  (2x  -  5y- z){3x +  Sy-\-2zy 


§  286;   page  266. 
4.   (2x  +  5)(2x  +  3).     5.   (3x  -  2)(3x  -  4).     6.   (4x  +  7)(4x  -  3). 

7.  (x+l  +  2\/3)(x  +  l-2>/3).  11.   (5X  +  3+ V3)(5x  +  3-\/3). 

8.  (2x+H-V2)(2x+l-\/2).        12.   (2\/2-2  +  3x)(2v/2  +  2-3x). 

9.  (6x  +  6)(Cx- 1).  13.   (7x  +  C)(7x  +  2). 
10.   (x  +  2)(4x-3).  14.  (l+8x)(5-2x). 

§  287;  page  267. 

4.   (x2  +  2x  +  3)(x2-2x  +  3).  5.  (x2  +  3x  -  5)(x2  -  3x  -  6). 

6.  (2a2  +  3a6  +  462)(2a2-3a6  +  462). 

7.  (3x2+ 4x?/- 2?/2)(3x2-4xy-2y^). 

8.  (4  m"'  +  3  mn  +  n'^){i  m2  -  Smn  +  n^). 

9.  (2a2  +  6a-7)(2a2-5a-7). 


84  ALGEBRA. 

10.  (3a;2  +  a;Vl3  +  3)(3a;2-a;\/l3  +  3). 

^  11.  (2»i2  +  mV5-2)(2m2-??iV5-2). 

12.  (x2  +  2  a; V2  +  4)  (x'^  -  2  xV2  +  4). 

13.  (_x^  +  xy/3-l){x^-xVS-l). 

14.  (3a2  +  5ax-d^2)(-3^2_5^^_5^2). 

15.  (4 a2  4-  am  +  6  m2) (4  a2  -  am  +  6  w2). 

16.  (6x2  +  X  -  2)(6x2  -  a;  -  2). 

17.  (5 m2  +  2  mx  +  4 x2) (5  ^2  -  2  mx  +  4 x2;. 

18.  (4x2 +  2x?/-72/2)(4x2-2x?/- 72/2).   - 

19.  (6a2-f  2a?>\/2-5  62)(6a2-2a&V2-6  62). 

§  288;   page  268. 

2.  V3±V32,  -V3±v:ri.  5    1  JrV^    -^^^^^. 

2  2 

3.  V3±V6,   -V3±V6.  6.   ^^1±2^,    -V3J:Vl5. 

4.  ±1,  ±i.  ,.  3vl±JivE2^ 

§  299;  page  277. 

3.  x  =  9,  y  =1;  x  =  6,  ?/  =  3;  x  =  3,  y  =  5. 

4.  X  =  4,  ?/  =  13  ;  X  =  8,  y  =  6.  5.    x 

6.  X  =  4,  y  =  122  ;  X  =  13,  y  =  91  ;  x  =  22,  ?/  =  60 

7.  X  =  3,  2/  =  50  ;  X  =  10,  y  =  26  ;  x  =  17,  y  =  2. 
9.    X  =  3,  ?/  =z  59  ;  X  =  13,  ?/  =  16. 

10.  X  =  78,  ?/  =  4  ;  X  =  59,  J/  =  12  ;  X  =  40,  ?/  =  20  ;  X  =  21,  y  =  28 ; 
x  =  2,  y  =  36.  II.    x  =  2,  y=l,  z  =  S. 

12.  X  =  2,  ?/  =  30,  0  =  3  ;  x  =  9,  y  =  IS,  z  =  48  ;  x  =  16,y  =  e,z  =  93. 

13.  x=2,  y  =  l.     14.  x=5,  i/=2.      15.  x=8,  y=6.     16.  x  =  3,  y=ll. 
17.    X  =  7,  ?/  =  1.  18.    X  =  9,  ?/  =  4. 

19.  Either  2  and  8,  or  6  and  3,  twenty-five  and  twenty-cent  pieces. 

20.  Either  1  and  17,  3  and  12,  5  and  7,  or  7  and  2,  fifty  and  twenty- 

19  2    10  7  1'  12 

cent  pieces.        21.   Either  —  and  -,  —  and  -,  or  -  and  — • 
9  5     9  5         9  5 

22.  Either  1,  18,  and  1;  4,  10,  and  6  ;  or  7,  2,  and  11,  half-dollars, 
quarter-dollars,  and  dimes.  23.    5  pigs,  10  sheep,  15  calves. 

24.  Either  17,  2,  and  8  ;  or  3,  11,  and  25,  quarter-dollars,  twenty- 
cent  pieces,  and  dimes. 


-3V2 

2 
±3V- 

■2 

2 

3,  y  = 

)0;  x  = 
8.  X 

5. 

:.31,  y  = 
=  3,  y 

:29. 

=  2. 

ANSWERS.  36 

§  322  ;   pages  285,  286. 

4.  8.       5.    30.       6.  -•     7.    If.       8.   ^^.      9.    10^.       10.    a:  -  3. 

32  a  +  o 

11.    2a- 1.  12.    -1,  ^-  13.    5,  22,  -4.  14.   ^. 

11  0 

15.  rK  =  ±a26,  ?/  =  ±a?)2.  16.  32,  18.  17.  25,  11.  18.  31,  17. 
19.  6,  8.  23.  3  :  4.  24.  a  :  -  6.  25.  1  or  -  15.  29.  5  :  4. 
30.   3:4.  31.    3,  9,  27. 

§  332 ;  pages  289,  290. 

3.  72.       4.   y  =  lz\       5.  J.       6.   ^.       7.   |     8.   -18.        9.  ]- 

o  V  o  4 

10.   579  ft.     11.  — ,    -  — .       12.    7.       13.    16.       14.  — •     15.  12  in. 
4  3x  2 

16.  3.     17.    5.      18.   9  in.     19.    15(>/3-l)  in,     20.   y=3  +  5x-4a;3. 

§  337;  page  292. 

2.    ;  =  69,  ;S'  =  432.     3.  Z  =  -77,  >S  =  -C30.     4.  Z  =  36,  6^  =  -  264. 

5.  ^^_60^   ^^_561.  e    ^^m    ^,^793. 

4  4  4  4 

7.    i=m,   ^=mi.  8.    Z  =  -21,  ^  =  -165. 

6  6  4 

9.    Z  =  -— ,   6^  =  -Zii.  10.    Z  =  34a+19&,  ^=162  a +  63  6. 

5  10 

11     ;_17?/-8a;    ^_80y-35a; 
~         2        '  2         ' 

§  338;  pages  294,  295. 

4.  a  =  l,  S=biO.  5.    a  =  7,  I  =  -69.  6.   d  =  3,  ^=552. 

7.    cZ  =  -  5,  Z  =  -  95.         8.    d  =  -,  n  =  35.        9.    a=-,  d  =  -—' 

4  5  15 

10.   Z  =  -,  n  =  16.  11.   n  =  22,  S  =  --  12.  a  =  -3,  Z  =  5. 

12  2 

13.    a  =  --,  n  =  9.         14.    «  =  §,<?  =  -!.  15.    df  =  --,  7i  =  13. 

3  2'  3  4' 

16.    d  =  -,  1  =  6.  17.    71  =  15,  Z  =  -  3  ;  or,  n  =  6,  I  =—- 

18.    «  =  -  -,  n  =  16  ;  or,  a  =  — ,  w  =  25.  19.    7i  =  16,  l=~  15. 

3  15 


36  ALGEBRA. 

21.   d  =  ^^^-  22.    d  =  ^(^-^^\  I  =  ^^-^^ 

w  —  1  w(n  —  1)  w 

23  ^^2>y-w(n-l)d    ^^2^+n(n-l)d 

2n  '  2  n 

24  w,  =  ^-^  +  ^    s  =  G  +  <^)(^-«  +  ^) 

f?        '  2  c? 

25.  «  =  ;-(n-l)(^,  .S'  =  ^[2?-(n-l)d]. 

26.  a  =  ^-^^',d  =  ?M:::^. 

w  n{n  —  1) 


2  2S-a-i  a+! 


7.   d=- 

5 
4' 

§  340; 

page  296. 

^■h- 

2.    X2  4-49. 

3    4«2  4.i 
4  a-2  -  1 

OQ            cZ  ±  V(2  ?  +  cZ)2  -  8  (Z6'   ^  _2  I  -\-  d  ^  V(2l  +  d)^  -  SdS 
29.    a  = ^  ,n-  — 

§  339  ;  page  296. 


§  341;  pages  297  to  299. 

3.    6050.  4.    250500.  6.    -50.  6.    10,  2,   -6,   -14. 

7.    840.       8.    65x  +  52|/.       9.    3,  5,  7,  9.       10.    100.       11.    44550. 

12.    31.        13.   ^.     14.    -6,-2,  2,  6,  10  ;  or,  21,  ^,  ^,  -  ^,  -  ^. 

15.    «wMl^.         16,    124.         17.    17.        18.    30.        19.    5.        20.    15. 

m  +  1 

21.    -3,  7,  17;  or,  _3,  -^,  -^.  22.    579. 

5         5 


§  345;  page  301. 
3.    Z=2ie7,  ;S'=3280.    4.  Z  =  ^,  >S'=^.    5.  Z=-1250,  ,9= -1042. 
6.    ^  =  2048,  *-4094.  T.    !  =  - J^,  «  =  -||. 


ANSWERS.  37 

8.    Z=-1280,  S  =  -^^'  9.  1  =  —,  S  =  -^^' 

2  625  625 

10.    ^  =  -243,  ^^_463.  ,,     ;=21  781. 

64'  192  128'  384 

12.    Z  =  768,  ^  =  2457. 
4 

§  346;   pages  302,  303. 

3.   a  =  1,^  =  511.         4.  a  =  3,  Z=-.         6.  r  =  -  4,  ,5  =  1638. 

6.   n  =  10,  S=—'  7.    a=-,  1  =  -- 

256  2'         2048 

8.    r  =  ?,  ^.191Il;or,r  =  -§,   S  =  '^.  9.  r  =  l,  n  =  9." 

2  384  2  384  2 

10.    lz=-  J-,  n  =  6.  11.  a  =  3,  w  =  7.  12.  r  =  -,  n  =  8. 

324  2 

13.    ;^«4-(r-l)>y,  j4    r  =  ^^^.  15.  a  =  rl-ir-l)S. 

r  S-l  ^         ^ 

16.    a  =  -L,S=   ^(r;-l).       n.a  =  ir-^)S        m-i^r-DS, 

yn-l  jrn-l(^y  _  J)  r«  —  1  »«"  —  1 


»•  -(-:y 


'  '^"TT — X" 


§  347  ;  pagre  304. 


2.  ?.  .       4.-1  6.    li.  8.    -^. 
2  6                                5                                     40 

3.  ^.  6.    -5.  7.   L2.  9.    A. 

6  55  21 


348;  page  305. 

25  ^     581  ^     107  «    2284 

2475* 


2.   1-.         3.   A.         4.   25.  5.    581,  107. 

-    11  27  36  990  925 


§  349  ;  page  305. 
2.  r=3.    3.  r=-2.     4.  r=±2.     b.r=±~.    6.  r=-4.    7.  r=±-. 

'Z  o 

§  350  ;   page  306. 
1.   2^.  2.  1.  3.  rt2  _  52.  4.   ^Ltll. 


X  —  '2y 


38  ALGEBRA. 

§351;   pages  306,  307. 

2.    -4.         3.  4,  12,  36,  108.        4.  5,  -10,  20;  or,  -5,  -10,  -20. 

5.    $4118.  6.  32  ft.  7.   -  ^^.  8.  (a'"b)"^. 

9.  -3,4, 11;  or,  13,  4, -5.     10.  A,  $108;  B,  $144;  C,  $192;  D,|256. 

11.    -  4,  1,  6,  36  ;  or,  8,  1,  -  6,  36.  12.   3. 

iQ     A    a   Q     ^v    76   190  475 

13.    4,  b,  9     or,  — , , • 

'    '     '       '  39    39'  39 

§  355  ;  page  309. 


--i 

•       -i, 

5. 

1.             6.    -A. 
61                         17 

7. 

1 

».  .  f . 

10, 

-10, 

10         2         1^ 

--,    -2,    --,    - 

10 
'  9* 

-  -1  - 

4 

-^j   ~ 

-4,  2,  4,1,  A,  2. 

'52    11    7 

0        4 

3        12 

2 

12 

3          4          6 

12 

1 

.11.    — , 

5 

~5'   "^'    ■ 

5' 

~35' 

10'       15'       25' 

55' 

5* 

.1.    4. 

-  -I 

12.  1-^^ 

X 

13. 

xy 
2x-y 

xy           xy 

14. 

5  and 

—3. 

Sx-2y  4x-Sy 

§  360 ;  page  314. 

10.  ai^  +  5  a^b^c  +  10  a%^c^  +  10  a'^h^c^  +  5  a^&i^c*  +  b^h^. 

11.  X^2m^(J  ^I0m^3n_|_  I5  x8"'2/6n  +  20  a;'5'»2/9''+  15  a;4'»|/12n  +  6  iC^'^J/l^w -f  ^/18«. 

12.  16  «4  _  32  a-3  +  24  a2  -  8  a  +  1. 

13.  x5+ 10a;*4-40x3  +  80a;2  4.80x  +  32. 

14.  «4  -  12  a36  +  54  a25-2  _  iqb  a63  +  81  &*. 

15.  1  +  12  m2  +  60  m*  +  160  m^  +  240  jji^  +  192  m^o  +  64  «ii2. 

17.  a;4  -f  5  x'="  +  10  o;^  +  10  x~^  +  5  a;""^"  +  x-^ 

18.  a^  -  14  a*^  +  84  J  -  280  a^  +  560  a^  -  672  a  +  448  a^  -  128. 

19.  243  +  405  x3  +  270  x^  +  90  iK^  +  15  x^^  +  x^^ 

20.  ?>r^  +  6  w"r^  +  15  9W~3  +  20  TO*  +  15  rn^'  +  6  w^^  +  mK 

21.  256  a^  -  256  a  'x-^  +  96  a^x^  -  16  a^x  +  xl     , 

22.  X-IO  -  f  X-8  y^  +  li>  X-6^8  _  |0  ^-4^12  _^  _5_.  a.-2yl6  _  _1  ^  2,20. 

23.  to12  +  20  m^-^  +  150  m^x-"^  +  500  w'^x-^  +  625  x-12. 


ANSWERS.  39 

24.  16  a^  +  16  a-  +  C  a"^'  4  a~^  +  ^^  a-^. 

25.  x^'  -  7  aj's^y"^  +  21  a;2i/"^  -  35  x^y'^^  +  35  x^y-i  -  21  x^?/"^ 

2    _3  _7 

+  lx''>y  2  -  y  ■*. 

26.  16  a~*  -  32  a-^h^  +  24  qTH  -  8  a~3-62  +  b'K  / 

_15  3        -?^   '^        _3   9        _3   J  ?.  / 

27.  3^2  ^  *  —  A"  ^"^wi'  +  I  x  ^m"^  —  I X  ^wi^  +  t  ic  ''w*  ^  —  m^.         \ 

28.  ac  +  16  a^  +  96  a'^'"  +  256  a^  +  256  ai 

29.  «3  _  18  a  V^  +  135  a'^x'^  -  640  a  V*  +  1215  ax~'^'  -  1458  ah~''^ 

+  729X-8. 

30.  32  «'^  -  240  a'^b  +  720  a^b'^  -  1080  a'-^ft^  4-  8 10  «&*  -  243  b^. 

31.  a 'ft's  +  7  a'ft"^  +  21  a^6-i  +  35  ah~^'  +  35  a'h^  +  21  a'^ft 

+  7  a~^6^  H-  a"^6i 

32.  81  m2n-2  -  216  mu"!  +  216  -  96  m-^n  +  16  m-^n^. 

34.  1  -4a;  +  10x2-  16a;3+  19x*-  16x5+  10 x^  _  4 a;7  +  a:8. 

35.  x8  +  4x7  +  14x«  + 28x6  4- 49x4 +  56x3  + 56x2 +  32X  +  16. 

36.  1+  12x  + 50x2 +  72x3 -21x4-  72x5  +  50x8  -  12x7  +  x^. 

37.  x8- 8x7 +  12x6 +  40x6 -74x4-  120x3  +  108x2  +  216 x  +  81. 

38.  1 +  5x+5x2-10x3-15x4+ 11x5+ 15x6-10  x7-5x8+5x9-xio. 

39.  xio  -  5x9  +  20x8  -  50x7  +  105 x^  -  161x5  +  210x4  -  200x3 

+  160x2 -80x  + 32. 

§  362  ;  page  316. 

2.  56a5x3.                     7.    Vis'a"^^*-  12.  j%\\a^h-^ 

3.  165  m3.                      8.    -220xi5y-3.  13.  42240  a'^x's". 

4.  126a564.                    9.    5005  a6»'+9''.  14.  21840  ^jAV^. 

5.  -11440x9.              10.    -  219648 x-6i/i  15.  - -^002  ^-'iy-i-^ 

6.  495w8n24.               11.   61236 rt""j\25.  16.  -i^^a^xu^ 

§  371;  page  323. 

3.  1 +4x-4x2  +  4x3-4xt+ .... 

4.  3  +  10x  +  40x-+ 160x''  +  640x4+ .... 
5.2  +  13x2  +  39x4+117x6  +  351x8+.... 

6.  2  X  -  I  x3  +  -2^t  x5  -  3S  x7  +  -IjV  ^^ . 

7.  1 +  x  +  x'J  +  2x4  +  5x5+ .... 


40  ALGEBKA. 

8.  2x-7x2  +  38a;3_204a:4  + 1096a;5 . 

9.  ix-2  +  |x-i  +  |f +  w^  +  ili^'  +  •.•. 
.         10.  i-ix-Y^-'-ffa:^  +  Ma^'+-- 

11.  l-2x  +  a;2  +  2x3-3x4+ -. 

12.  2  +  9x4-23x2  +  47x'5  +  73a:t+ .... 

13.  x-3  +  5x-2  +  20X-1  +  106  +  570x  +  .... 

14.  3x-2  +  14x-i  +  39  +  lOlx  +  264x2  +  .... 

15.  ^x2-2x3  +  fx*-|x5  +  |x6+.... 
ID.  3  +  eX—  2yX    —  TT^  2¥3^    —  •••. 
17.  fX-1-  fX+^x2-h  3x3-  |X4+  .... 

§  372  ;  page  324. 

2.  1 +2x-2x2  +  4x3- 10x*+ .... 

3.  l_fx-^/-x2-Jj-V-x3--3ji^/-x*-.... 

4.  1  +X-X2  +  X3-  fx4+  .... 

5.  l-ix-fx^-J^jX^-rVsX*-.... 

6.  l+x-x2  +  fx5-J3'>xi  + .... 

7.  1 -ix  +  fx2 +  i!x3  + ^13X^4- -. 

§  374 ;  page  325. 

4.   J- ? 6.   »         1  1 


8 

4                5- 

4. 

2x  +  3      2x-3 

6. 

8                7 

7. 

2x  +  3      3x-2 

0. 

.-IN 

3x     3(5x-6)  X     x+5     x-6 

4a  3a  g        10  3 


x  +  5a     X  — a  2  — 5x     4 -f  x 

1  10.    1_^_+     2  1 


2(2x-l)      2(4x-3)      3x  +  2  x     x-2     x  +  3     x-3 


§  376  ;   page  327. 

9  e_       2  1 


2x-3      (2x-3)2  6(5x+2)     (6x+2)2    5(5x+2)3 


1 g  4  „       1  4  3 

x  +  5      (X  +  6)2      (X  +  5)3' 


^13  5 


3x-l     (3x-l)2     (3x-l)3         x+2     (x  +  2)2     (x+2)3     (x  +  2)4 


6._2_+_i2 _1 92,5  4 


2x-3     (2x-3)2     (2x-3)3  3(3x-2)     3(3x-2)2     3(3x-2)* 


ANSWERS.  41 

§  377 ;  pagre  328. 

2  2___5 8_.  6J+2__3 i_. 

X     ic-3      (x-3)2  X     x2     x-l      (x-l)« 

3  3      4       2    ^       5  g    1 3 5 

'xx2x3x  +  4  'xx+1      (x  +  2)2* 

4.  --^_  +  -^^ I 7.-1 L_+ ^. 

3x-l      2x  +  3      (2x  +  3)-^         4x4-1     2(2x-3)     2(2x-3)« 

§  378 ;  page  329. 

2.3x-2+-«---A_-.        4.x-l-l-l+4-  +  ^!_. 
x  +  2     3x-l  X     x2     x8     x+l 

3.  2 ^  + — -•  5.  x4-2+?--l ?— + 2 

X  -  2      (X  -  2)8  X     x^     X  -  1      (X  -  1)2 

6.  x2  +  3-3..1+2_.^_. 
X     x2     x3     x+3 

§  379 ;  page  330. 

2  -2,-.  I     S  a;  -  1  ,  g        3  _         1          X  - 1 

'   x  +  i     a;2-x  +  r  '  x  +  1     x-1     x^+l' 

3  5        ^      2x  +  3  g  _J 3X4-.1 

3  X  +  1     X-  -  X  +  3*  ■  2  X  -  3     4  x2  +  6  X  +  9* 

4  4      _  x-3  ^      .5x  +  n     _    3x-4 

■   2  X  -  5     x2  +  2*  '  x^-^x+l     x2  -  X  4- 1' 

§  380  ;  pagr6  331. 

2.  x  =  y  +  ?/2  +  y3^y4+  ....  6.  a;  =  y+  i  ?/2  4.  i  ^3+ i  y4_|.  .... 

3.  x  =  .v+i2/2+^2/»+ Ay^+-.    7.  x  =  2y-2t/2  4-4y3_|y4+.... 

4.  X  =  y-2y2  4.5y8_i4y4+....    g.  .«  =  y  _  yS  4.  ^6  _  ^,7  ^  .... 

6.  x  =  2/  +  3y2  +  13y3  +  67t/44--.    9.  a- =  .v-i  y3  +  ^^y6_^Y^y74..... 

§  383 ;  page  336. 
7.  J  -\  a~^x  -  /j  a"^x2  -  ^1^  a~'^''ot^  -  ^^l-g  a~'^'x* . 

9.  a-6  +  6  a-76  +  21  a-»b^  +  56 a-^fts  +  126  a-^%*+  .... 

10.  x^  -  5x!/  +  V-a^V  -  t  a;"V  _  5  3.-1,/*+  .... 

11.  m8  -  f  w^w"^  +  Jg<i»ni2»r3  -  f^»n"?r-'  4-  |f  m^«n"^ . 


42 


ALGEBRA. 


12. 
13. 

14. 

15. 

16. 

17. 

18. 

2. 
3. 
4. 
5. 
6. 


a-l  +  iOf-^iC    2  +  I  «-%-!+  [|  a-13a;-2  ^   1.9|  (j-17^-2_|.  ..., 

x-2  -  4  x-^y  +  16  x-6y2  _  64  x^^y^  +  256  x'^V • 

x~'^'  -\-lx~  ^yz  +  -3/-  a:"^2/2;s2  ^  3.5  a;"2?/3^3  +  3^5  x^y*z*-{-  .... 

m~2  +  lOm-3/i"^  +  60?»~^«"^  +  280  ?^r*n-2  +  1120m~^w~^+  .... 

_3     3  _11     11  _19     19  _27     27  _35     35 

a  ^b^-la   ib^^^a  -^h^-^J^a   ffti+Hlf  a  ^h'^  -.... 

8     3  .113  14     9  17 

X  4-  bx^y^  +  20a;"^>2  +  ^io^c^-^t  ^  7103.3-^3+  .... 

_9  _3     2  34  96  158 


^i^oT'x^ 

_1  3 
23  1     ff    "27)6 

1365  a;ii. 
-192a:7yl 

tV-t  a~%8. 


§  384 ;   pagre  337. 

7.  ^Wx\ 

8.  - 


_4J) 


9.    -  2002  a:-i6m«. 
10.   ^p  wr'^8^n--8. 


12.    Y#-«""'^'ft-^ 


13.      -l-4_8^ 


14.   220x-iV2>-6 
15. 


_2  3 


11*      3 2V6  8  ^ 


^-:»^«. 


§  385 ;  page  338. 

5.09902.                              4.  2.08008. 

9.89949.                              5.  2.97182. 

; 

§  397 ;  page  342. 

1.5441.                  7.    2.1003.  12.    2.5104. 

1.6990.                  8.    2.2922.  13.    2.5774. 

1.6282.                  9.   2.3892.  14.    2.6074. 

1.8751.                10.    2.3222.  15.    2.9421. 

1.6020.               11.    2.7960.  16.   2.8363. 


6.  2.03055. 

7.  1.96100. 


17.  3.0512. 

18.  3.4192. 

19.  3.7814. 

20.  4.0794. 

21.  4.2006. 


2.  .5229. 

3.  .2431. 

4.  1.1549. 


§  309: 

5.  1.6532. 

6.  .2589. 

7.  2.3522. 


page  343. 

8.  .2831. 

9.  .7939. 
10.   2.1303. 


11.  1.4592. 

12.  1.3468. 

13.  2.0424. 


§  402  ;  page  344, 

3.  3. .3397.  5.    .7525.  7.    7.7205. 

4.  4.19-10.  6.    .6338.  8.    .4824. 


10. 


.286.3. 
1.0460. 


ANSWERS. 


4S 


11.  .3943. 

12.  .0682. 

13.  .1165. 

14.  .0939. 


2.  0.4471 

3.  1.0491. 

4.  9.7993-10. 


15.  .4042. 

16.  .6250. 

17.  .4978. 

18.  .2542. 


20.  .0495. 

21.  .0366, 

22.  .7007. 

23.  .8752. 


24.  .0794.     -^ 

25.  .4248. 

26.  .1341. 

27.  .1807. 


§  406  ;  page  346. 

6.  1.5104.  10.    6.5353  -  10.  14.  3.2646. 

7.  7.5741-10.    11.    9.9421-10.  16.  0.1151. 

8.  3.8293.  12.   0.4134.  16.  0.7335. 


5.  8.9912  -  10.     9.    8.5932  -  10.    13.    2.4383. 

§  411 ;  pagre  350. 

6.  3.0286.              9.    7.8605-10.    12.    2.4032.  15.  7.8108-10. 

7.  1.9189.            10.    0.8923.            13.   9.9632-10.  16.  8.1332-10. 

8.  9.9830-10.  11.    6.5783-10.    14.    3.6099.  17.  0.6059. 

§  413 ;  page  351. 


4. 

64.26. 

7. 

.8143. 

10. 

.09215. 

13.  .5061. 

5. 

2273. 

8. 

.004897. 

11. 

64.23. 

14.  366.8. 

6. 

461.2. 

9. 

7.488. 

12. 

.003856. 

15.  17008. 

16.  .0001994, 

§ 

418;  pages 

355, 

,  356. 

1. 

189.7. 

15. 

-1.167. 

29. 

.6682. 

45. 

2.627. 

2. 

8.243. 

16. 

-.002893. 

30. 

.6458. 

46. 

2.527. 

3. 

-  1933. 

17. 

3692. 

31. 

.1377. 

47. 

-.8378. 

4. 

.3091. 

18. 

.2777. 

32. 

-.3702. 

48. 

1.033. 

5. 

.002976. 

19. 

-  16893. 

35. 

30.12. 

49. 

.2984. 

6. 

-.01213. 

20. 

.001233. 

36. 

2.487. 

50. 

.3697. 

7. 

6.359. 

21. 

316.2. 

37. 

1.056. 

61. 

.7945. 

8. 

.03018. 

22. 

.7652. 

38. 

.0006777. 

52. 

.9348. 

9. 

-  6.853. 

23. 

243.9. 

39. 

.007105. 

53. 

179.6. 

10. 

311.9. 

24. 

.00001085 

40. 

.8335. 

64. 

1.883. 

11. 

.2239. 

25. 

2.236. 

41. 

.5428. 

55. 

.0001931, 

12. 

-.009544. 

26. 

1.149. 

42. 

-  36.03. 

56. 

-.09954. 

13. 

.1261. 

27. 

-  1.276. 

43. 

-  11.11. 

67. 

.1711. 

14. 

.02367. 

28. 

1.778. 

44. 

.9432. 

58. 

-  74.88. 

44 


ALGEBRA. 


S.    .28301. 


§  419  ;  page  357. 
4.    -2.172.  5.    1.155.  6.    -.1766. 


51oge 


3  lag  a 


log  a  —  2  log  b 


log  n  —  4  log  m 


9. 


10.   4,  -  1. 


11.    ^^log^-loga_^i,  12.    ^^lo^[(r-l)>y+a]-log«, 

log  r  log  r 


II.    n  = 


log  ?  —  log  g 


log(6^-a)-log(>S-0 


4-1. 


14.    ^^log?-log[W-(r-l)^j^^^ 
logr 

•     §420;  pai^e  358. 

2.   3.701.  3.    -.06552.  4.    -2.761. 


5.    2.389. 


6.    -.3763.         7.    .3731.         9.    4.  10.   -•         11.    --•         12.    -• 

3  3  5 


OF  THE 

UNIVERSITY 


OVERDUE. 


LD2l-l00m-7,'40  (6936s) 


